M09 Polynomial and Rational Inequalities.pdf

Inequalities in One

Variable

At the end of this lecture, a student must be able to:

Solve precisely for the solution set of an inequality.

Give the correspondence between inequalities and intervals on the real number line

Solve inequalities involving polynomials and rational expressions using a Table of Signs

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Inequalities

An

inequality

is a statement saying that one

expression is less than or equal to another.

3x−2<3x+ 1

1−x≤ 2x

x+ 1

x+ 3≥x+ 7

A

solution

to an inequality is a value of the

variable that makes the inequality true.

The

solution set

of an inequality is the set

of all solutions of the inequality.

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Inequalities

An

absolute inequality

is an inequality that

is true for all permissible values.

Example: x2+ 1 ≥2x x <|x|+ 1

A

conditional inequality

is an inequality

that is true only for some real values.

Example: x >2 1−x≤ 2x

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Interval Notation

All real numbers can be putin sequenceon a number line.

−5 −4 −3 −2 −1 0 1 2 3 4 5

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Interval Notation

Leta∈_R.

−5 −4 −3 −2 −1 0 1 2 3 4 5

a

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Interval Notation

Leta∈_R.

a

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Interval Notation

Leta∈_R.

−5 −4 −3 −2 −1 0 1 2 3 4 5

a

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Interval Notation

Leta∈_R.

a

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Interval Notation

Leta∈_R.

−5 −4 −3 −2 −1 0 1 2 3 4 5

a

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Interval Notation

Leta, b∈_R such that a < b.

a b

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Interval Notation

Leta, b∈_R such that a < b.

a b

Interval Notation

Leta, b∈_R such that a < b.

a b

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Interval Notation

Leta, b∈_R such that a < b.

a b

Interval Notation

Leta, b∈_R such that a < b.

a b

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Order Axioms

Let

a, b

∈

R

such that

a < b

. Then

•

a

+

c < b

+

c

for all

c

∈

R

•

if

c >

0, then

ac < bc

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Linear Inequalities

Inequalities that can be written in the form

ax

+

b >

0

.

(here,>may be replaced by<,≥,≤)

To solve these inequalities, isolatex on one side using the order axioms.

Solution:

3x+ 1 > x+ 7 given

3x−x > 7−1 additive property 2x > 6

x > 3

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Linear Inequalities

Inequalities that can be written in the form

ax

+

b >

0

.

(here,>may be replaced by<,≥,≤)

To solve these inequalities, isolatex on one side using the order axioms.

Example: Find the solution set of 3x+ 1> x+ 7. Solution:

3x+ 1 > x+ 7 given

3x−x > 7−1 additive property 2x > 6

x > 3

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Linear Inequalities

Inequalities that can be written in the form

ax

+

b >

0

.

(here,>may be replaced by<,≥,≤)

To solve these inequalities, isolatex on one side using the order axioms.

Example: Find the solution set of 3x+ 1> x+ 7. Solution:

3x+ 1 > x+ 7 given

3x−x > 7−1 additive property

x > 3

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Linear Inequalities

Inequalities that can be written in the form

ax

+

b >

0

.

(here,>may be replaced by<,≥,≤)

To solve these inequalities, isolatex on one side using the order axioms.

Example: Find the solution set of 3x+ 1> x+ 7. Solution:

3x+ 1 > x+ 7 given

3x−x > 7−1 additive property 2x > 6

x > 3

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Linear Inequalities

Inequalities that can be written in the form

ax

+

b >

0

.

(here,>may be replaced by<,≥,≤)

To solve these inequalities, isolatex on one side using the order axioms.

Example: Find the solution set of 3x+ 1> x+ 7. Solution:

3x+ 1 > x+ 7 given

3x−x > 7−1 additive property 2x > 6

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Linear Inequalities

Inequalities that can be written in the form

ax

+

b >

0

.

(here,>may be replaced by<,≥,≤)

To solve these inequalities, isolatex on one side using the order axioms.

Example: Find the solution set of 3x+ 1> x+ 7. Solution:

3x+ 1 > x+ 7 given

3x−x > 7−1 additive property 2x > 6

x > 3

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

x∈(a, b) ⇔ a < x < b

Example: Find the solution set of 2<5−3x <11. Solution:

2 < 5−3x < 11

(₋1₃)(

−3

)

< −3x

(₋1₃)

< 6

(₋1₃)

1

>

x

>

−2

m

−2 < x < 1

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

x∈(a, b) ⇔ a < x < b

Example: Find the solution set of 2<5−3x <11. Solution:

2

−5

< 5−3x

−5

< 11

−5

(₋1₃)(

−3

)

< −3x

(₋1₃)

< 6

(₋1₃)

1

>

x

>

−2

m

−2 < x < 1

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

x∈(a, b) ⇔ a < x < b

Example: Find the solution set of 2<5−3x <11. Solution:

2−5 < 5−3x−5 < 11−5

(₋1₃)(

−3

)

<

−3x

(₋₃)

< 6

(₋₃)

1

>

x

>

−2

m

−2 < x < 1

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

x∈(a, b) ⇔ a < x < b

Example: Find the solution set of 2<5−3x <11. Solution:

2−5 < 5−3x−5 < 11−5

(₋1₃)(

−3

)

< −3x

(₋1₃) < 6

(₋1₃)

1

>

x

>

−2

m

−2 < x < 1

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

x∈(a, b) ⇔ a < x < b

Example: Find the solution set of 2<5−3x <11. Solution:

2−5 < 5−3x−5 < 11−5

(₋1₃)(

−3

)

< −3x

(₋1₃)

< 6

(₋1₃)

1 x −2

m

−2 < x < 1

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

x∈(a, b) ⇔ a < x < b

Example: Find the solution set of 2<5−3x <11. Solution:

2

−5

< 5−3x

−5

< 11

−5 (₋1₃)(

−3

)

< −3x

(₋1₃)

< 6

(₋1₃) 1

>

x

>

−2

m

−2 < x < 1

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

x∈(a, b) ⇔ a < x < b

Example: Find the solution set of 2<5−3x <11. Solution:

2

−5

< 5−3x

−5

< 11

−5

(₋1₃)(−3) < −3x(₋1₃) < 6(₋1₃) 1

> x −2

m

−2 < x < 1

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

x∈(a, b) ⇔ a < x < b

Example: Find the solution set of 2<5−3x <11. Solution:

2

−5

< 5−3x

−5

< 11

−5

(₋1₃)(−3) < −3x(₋1₃) < 6(₋1₃) 1

>

x

> −2

m

−2 < x < 1

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

x∈(a, b) ⇔ a < x < b

Example: Find the solution set of 2<5−3x <11. Solution:

2 < 5−3x < 11

(₋1₃)(−3) < −3x(₋1₃) < 6(₋1₃)

1 > x > −2

m

−2 < x < 1

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

x∈(a, b) ⇔ a < x < b

Example: Find the solution set of 2<5−3x <11. Solution:

2

−5

< 5−3x

−5

< 11

−5 (₋1₃)(

−3

)

< −3x

(₋1₃)

< 6

(₋1₃)

1 > x > −2

m

−2 < x < 1

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

x∈(a, b) ⇔ a < x < b

Example: Find the solution set of 2<5−3x <11. Solution:

2 < 5−3x < 11

(₋1₃)(

−3

)

< −3x

(₋1₃)

< 6

(₋1₃)

1 > x > −2

m

−2 < x < 1

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

x∈(a, b) ⇔ a < x < b

Example: Find the solution set of 2<5−3x <11. Solution:

2

−5

< 5−3x

−5

< 11

−5 (₋1₃)(

−3

)

< −3x

(₋1₃)

< 6

(₋1₃)

1 > x > −2

m

−2 < x < 1

Polynomial and Rational Inequalities

To solve:

1. Write the inequality in the formA >0, A <0, A≥0 orA≤0. (standard form)

2. Find all real values ofxfor which the numerator or the denominator is zero. (critical points)

3. Make a table describing the sign of each factor in the numerator and the denominator on each subinterval into which the critical points subdivide_R. (table of signs)

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find the solution set ofx2_{−2x <2x−3.}

Solution:

x2−4x+ 3 < 0 Standard Form (x−1)(x−3) < 0 Factor

critical numbers : 1,3

Table of Signs:

(−∞,1) (1,3) (3,∞) x−1

− + +

x−3 − − +

(x−1)(x−3) + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find the solution set ofx2_{−2x <2x−3.}

Solution:

x2−4x+ 3 < 0 Standard Form

(x−1)(x−3) < 0 Factor critical numbers : 1,3

Table of Signs:

(−∞,1) (1,3) (3,∞) x−1

x−3 − − +

(x−1)(x−3) + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find the solution set ofx2_{−2x <2x−3.}

Solution:

x2−4x+ 3 < 0 Standard Form (x−1)(x−3) < 0 Factor

critical numbers : 1,3

Table of Signs:

(−∞,1) (1,3) (3,∞) x−1

− + +

x−3 − − +

(x−1)(x−3) + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find the solution set ofx2_{−2x <2x−3.}

Solution:

x2−4x+ 3 < 0 Standard Form (x−1)(x−3) < 0 Factor

critical numbers : 1,3

Table of Signs:

(−∞,1) (1,3) (3,∞) x−1

x−3 − − +

(x−1)(x−3) + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find the solution set ofx2_{−2x <2x−3.}

Solution:

x2−4x+ 3 < 0 Standard Form (x−1)(x−3) < 0 Factor

critical numbers : 1,3

Table of Signs:

(−∞,1) (1,3) (3,∞) x−1

− + +

x−3 − − +

(x−1)(x−3) + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find the solution set ofx2_{−2x <2x−3.}

Solution:

x2−4x+ 3 < 0 Standard Form (x−1)(x−3) < 0 Factor

critical numbers : 1,3

Table of Signs:

(−∞,1) (1,3) (3,∞) x−1

x−3

− − +

(x−1)(x−3)

+ − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find the solution set ofx2_{−2x <2x−3.}

Solution:

x2−4x+ 3 < 0 Standard Form (x−1)(x−3) < 0 Factor

critical numbers : 1,3

Table of Signs:

(−∞,1) (1,3) (3,∞)

x−1 −

+ +

x−3

− − +

(x−1)(x−3)

+ − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find the solution set ofx2_{−2x <2x−3.}

Solution:

x2−4x+ 3 < 0 Standard Form (x−1)(x−3) < 0 Factor

critical numbers : 1,3

Table of Signs:

(−∞,1) (1,3) (3,∞)

x−1 − +

x−3

− − +

(x−1)(x−3)

+ − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find the solution set ofx2_{−2x <2x−3.}

Solution:

x2−4x+ 3 < 0 Standard Form (x−1)(x−3) < 0 Factor

critical numbers : 1,3

Table of Signs:

(−∞,1) (1,3) (3,∞)

x−1 − + +

x−3

− − +

(x−1)(x−3)

+ − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find the solution set ofx2_{−2x <2x−3.}

Solution:

x2−4x+ 3 < 0 Standard Form (x−1)(x−3) < 0 Factor

critical numbers : 1,3

Table of Signs:

(−∞,1) (1,3) (3,∞)

x−1 − + +

x−3 −

(x−1)(x−3)

+ − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find the solution set ofx2_{−2x <2x−3.}

Solution:

x2−4x+ 3 < 0 Standard Form (x−1)(x−3) < 0 Factor

critical numbers : 1,3

Table of Signs:

(−∞,1) (1,3) (3,∞)

x−1 − + +

x−3 − −

+

(x−1)(x−3)

+ − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find the solution set ofx2_{−2x <2x−3.}

Solution:

x2−4x+ 3 < 0 Standard Form (x−1)(x−3) < 0 Factor

critical numbers : 1,3

Table of Signs:

(−∞,1) (1,3) (3,∞)

x−1 − + +

x−3 − − +

(x−1)(x−3)

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find the solution set ofx2_{−2x <2x−3.}

Solution:

x2−4x+ 3 < 0 Standard Form (x−1)(x−3) < 0 Factor

critical numbers : 1,3

Table of Signs:

(−∞,1) (1,3) (3,∞)

x−1 − + +

x−3 − − +

(x−1)(x−3) +

− +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find the solution set ofx2_{−2x <2x−3.}

Solution:

x2−4x+ 3 < 0 Standard Form (x−1)(x−3) < 0 Factor

critical numbers : 1,3

Table of Signs:

(−∞,1) (1,3) (3,∞)

x−1 − + +

x−3 − − +

(x−1)(x−3) + −

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find the solution set ofx2_{−2x <2x−3.}

Solution:

x2−4x+ 3 < 0 Standard Form (x−1)(x−3) < 0 Factor

critical numbers : 1,3

Table of Signs:

(−∞,1) (1,3) (3,∞)

x−1 − + +

x−3 − − +

(x−1)(x−3) + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find the solution set ofx2_{−2x <2x−3.}

Solution:

x2−4x+ 3 < 0 Standard Form (x−1)(x−3) < 0 Factor

critical numbers : 1,3

Table of Signs:

(−∞,1) (1,3) (3,∞)

x−1 − + +

x−3 − − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find the solution set ofx2_{−2x <2x−3.}

Solution:

x2−4x+ 3 < 0 Standard Form (x−1)(x−3) < 0 Factor

critical numbers : 1,3

Table of Signs:

(−∞,1) (1,3) (3,∞)

x−1 − + +

x−3 − − +

(x−1)(x−3) + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x satisfying x+ 2

x ≤x

0 ≤ x−x+ 2

x Standard Form

0 ≤ x

2 _−x−2

x Simplify

0 ≤ (x+ 1)(x−2)

x Factor

critical numbers : −1,0,2

Table of Signs: (−∞,−1) (−1,0) (0,2) (2,∞) x+ 1

x

− − + +

x−2

− − − +

x2_−x−2

x

− + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x satisfying x+ 2

x ≤x

0 ≤ x−x+ 2

x Standard Form

0 ≤ x

2 _−x−2

x Simplify

0 ≤ (x+ 1)(x−2)

x Factor

critical numbers : −1,0,2

Table of Signs: (−∞,−1) (−1,0) (0,2) (2,∞) x+ 1

− + + +

x

− − + +

x−2

− − − +

x2_−x−2

x

− + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x satisfying x+ 2

x ≤x

0 ≤ x−x+ 2

x Standard Form

0 ≤ x

2 _−x−2

x Simplify

0 ≤ (x+ 1)(x−2)

x Factor

critical numbers : −1,0,2

Table of Signs: (−∞,−1) (−1,0) (0,2) (2,∞) x+ 1

x

− − + +

x−2

− − − +

x2_−x−2

x

− + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x satisfying x+ 2

x ≤x

0 ≤ x−x+ 2

x Standard Form

0 ≤ x

2 _−x−2

x Simplify

0 ≤ (x+ 1)(x−2)

x Factor

critical numbers : −1,0,2

Table of Signs: (−∞,−1) (−1,0) (0,2) (2,∞) x+ 1

− + + +

x

− − + +

x−2

− − − +

x2_−x−2

x

− + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x satisfying x+ 2

x ≤x

0 ≤ x−x+ 2

x Standard Form

0 ≤ x

2 _−x−2

x Simplify

0 ≤ (x+ 1)(x−2)

x Factor

critical numbers : −1,0,2

Table of Signs: (−∞,−1) (−1,0) (0,2) (2,∞) x+ 1

x

− − + +

x−2

− − − +

x2_−x−2

x

− + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x satisfying x+ 2

x ≤x

0 ≤ x−x+ 2

x Standard Form

0 ≤ x

2 _−x−2

x Simplify

0 ≤ (x+ 1)(x−2)

x Factor

critical numbers : −1,0,2

Table of Signs: (−∞,−1) (−1,0) (0,2) (2,∞) x+ 1

− + + +

x

− − + +

x−2

− − − +

x2_−x−2

x

− + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x satisfying x+ 2

x ≤x

0 ≤ x−x+ 2

x Standard Form

0 ≤ x

2 _−x−2

x Simplify

0 ≤ (x+ 1)(x−2)

x Factor

critical numbers : −1,0,2

Table of Signs: (−∞,−1) (−1,0) (0,2) (2,∞)

x+ 1 −

x

− − + +

x−2

− − − +

x2_−x−2

x

− + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x satisfying x+ 2

x ≤x

0 ≤ x−x+ 2

x Standard Form

0 ≤ x

2 _−x−2

x Simplify

0 ≤ (x+ 1)(x−2)

x Factor

critical numbers : −1,0,2

Table of Signs: (−∞,−1) (−1,0) (0,2) (2,∞)

x+ 1 − +

+ +

x

− − + +

x−2

− − − +

x2_−x−2

x

− + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x satisfying x+ 2

x ≤x

0 ≤ x−x+ 2

x Standard Form

0 ≤ x

2 _−x−2

x Simplify

0 ≤ (x+ 1)(x−2)

x Factor

critical numbers : −1,0,2

Table of Signs: (−∞,−1) (−1,0) (0,2) (2,∞)

x+ 1 − + +

x

− − + +

x−2

− − − +

x2_−x−2

x

− + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x satisfying x+ 2

x ≤x

0 ≤ x−x+ 2

x Standard Form

0 ≤ x

2 _−x−2

x Simplify

0 ≤ (x+ 1)(x−2)

x Factor

critical numbers : −1,0,2

Table of Signs: (−∞,−1) (−1,0) (0,2) (2,∞)

x+ 1 − + + +

x

− − + +

x−2

− − − +

x2_−x−2

x

− + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x satisfying x+ 2

x ≤x

0 ≤ x−x+ 2

x Standard Form

0 ≤ x

2 _−x−2

x Simplify

0 ≤ (x+ 1)(x−2)

x Factor

critical numbers : −1,0,2

Table of Signs: (−∞,−1) (−1,0) (0,2) (2,∞)

x+ 1 − + + +

x −

x−2

− − − +

x2_−x−2

x

− + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x satisfying x+ 2

x ≤x

0 ≤ x−x+ 2

x Standard Form

0 ≤ x

2 _−x−2

x Simplify

0 ≤ (x+ 1)(x−2)

x Factor

critical numbers : −1,0,2

Table of Signs: (−∞,−1) (−1,0) (0,2) (2,∞)

x+ 1 − + + +

x − −

+ +

x−2

− − − +

x2_−x−2

x

− + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x satisfying x+ 2

x ≤x

0 ≤ x−x+ 2

x Standard Form

0 ≤ x

2 _−x−2

x Simplify

0 ≤ (x+ 1)(x−2)

x Factor

critical numbers : −1,0,2

Table of Signs: (−∞,−1) (−1,0) (0,2) (2,∞)

x+ 1 − + + +

x − − +

x−2

− − − +

x2_−x−2

x

− + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x satisfying x+ 2

x ≤x

0 ≤ x−x+ 2

x Standard Form

0 ≤ x

2 _−x−2

x Simplify

0 ≤ (x+ 1)(x−2)

x Factor

critical numbers : −1,0,2

Table of Signs: (−∞,−1) (−1,0) (0,2) (2,∞)

x+ 1 − + + +

x − − + +

x−2

− − − +

x2_−x−2

x

− + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x satisfying x+ 2

x ≤x

0 ≤ x−x+ 2

x Standard Form

0 ≤ x

2 _−x−2

x Simplify

0 ≤ (x+ 1)(x−2)

x Factor

critical numbers : −1,0,2

Table of Signs: (−∞,−1) (−1,0) (0,2) (2,∞)

x+ 1 − + + +

x − − + +

x−2 −

x2_−x−2

x

− + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x satisfying x+ 2

x ≤x

0 ≤ x−x+ 2

x Standard Form

0 ≤ x

2 _−x−2

x Simplify

0 ≤ (x+ 1)(x−2)

x Factor

critical numbers : −1,0,2

Table of Signs: (−∞,−1) (−1,0) (0,2) (2,∞)

x+ 1 − + + +

x − − + +

x−2 − −

− +

x2_−x−2

x

− + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x satisfying x+ 2

x ≤x

0 ≤ x−x+ 2

x Standard Form

0 ≤ x

2 _−x−2

x Simplify

0 ≤ (x+ 1)(x−2)

x Factor

critical numbers : −1,0,2

Table of Signs: (−∞,−1) (−1,0) (0,2) (2,∞)

x+ 1 − + + +

x − − + +

x−2 − − −

x2_−x−2

x

− + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x satisfying x+ 2

x ≤x

0 ≤ x−x+ 2

x Standard Form

0 ≤ x

2 _−x−2

x Simplify

0 ≤ (x+ 1)(x−2)

x Factor

critical numbers : −1,0,2

Table of Signs: (−∞,−1) (−1,0) (0,2) (2,∞)

x+ 1 − + + +

x − − + +

x−2 − − − +

x2_−x−2

x

− + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x satisfying x+ 2

x ≤x

0 ≤ x−x+ 2

x Standard Form

0 ≤ x

2 _−x−2

x Simplify

0 ≤ (x+ 1)(x−2)

x Factor

critical numbers : −1,0,2

Table of Signs: (−∞,−1) (−1,0) (0,2) (2,∞)

x+ 1 − + + +

x − − + +

x−2 − − − +

x2_−x−2

x −

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x satisfying x+ 2

x ≤x

0 ≤ x−x+ 2

x Standard Form

0 ≤ x

2 _−x−2

x Simplify

0 ≤ (x+ 1)(x−2)

x Factor

critical numbers : −1,0,2

Table of Signs: (−∞,−1) (−1,0) (0,2) (2,∞)

x+ 1 − + + +

x − − + +

x−2 − − − +

x2_−x−2

x − +

− +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x satisfying x+ 2

x ≤x

0 ≤ x−x+ 2

x Standard Form

0 ≤ x

2 _−x−2

x Simplify

0 ≤ (x+ 1)(x−2)

x Factor

critical numbers : −1,0,2

Table of Signs: (−∞,−1) (−1,0) (0,2) (2,∞)

x+ 1 − + + +

x − − + +

x−2 − − − +

x2_−x−2

x − + −

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x satisfying x+ 2

x ≤x

0 ≤ x−x+ 2

x Standard Form

0 ≤ x

2 _−x−2

x Simplify

0 ≤ (x+ 1)(x−2)

x Factor

critical numbers : −1,0,2

Table of Signs: (−∞,−1) (−1,0) (0,2) (2,∞)

x+ 1 − + + +

x − − + +

x−2 − − − +

x2_−x−2

x − + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x satisfying x+ 2

x ≤x

0 ≤ x−x+ 2

x Standard Form

0 ≤ x

2 _−x−2

x Simplify

0 ≤ (x+ 1)(x−2)

x Factor

critical numbers : −1,0,2

Table of Signs: (−∞,−1) (−1,0) (0,2) (2,∞)

x+ 1 − + + +

x − − + +

x−2 − − − +

x2_−x−2

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x satisfying x+ 2

x ≤x

0 ≤ x−x+ 2

x Standard Form

0 ≤ x

2 _−x−2

x Simplify

0 ≤ (x+ 1)(x−2)

x Factor

critical numbers : −1,0,2

Table of Signs: (−∞,−1) (−1,0) (0,2) (2,∞)

x+ 1 − + + +

x − − + +

x−2 − − − +

x2_−x−2

x − + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Strategies in Solving Inequalities

•

Always put inequalities involving polynomials

or rational expressions in the standard form

E <

0 or

E >

0

.

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Strategies in Solving Inequalities

•

Always put inequalities involving polynomials

or rational expressions in the standard form

E <

0 or

E >

0

.

Inequalities with Absolute Value

Let

E

be any real expression and

k

∈

R

.

If

k <

0 (

k

is a negative number), then

•

|

E

|

> k

is

always

true

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Inequalities with Absolute Value

Let

E

be any real expression and let

k >

0. Then

|

E

|

< k

⇔ −

k < E < k

⇔

E >

−

k

and

E < k

|

E

| ≤

k

⇔ −

k

≤

E

≤

k

⇔

E

≥ −

k

and

E

≤

k

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Inequalities with Absolute Value

Let

E

be any real expression and let

k >

0. Then

|

E

|

< k

⇔ −

k < E < k

⇔

E >

−

k

and

E < k

|

E

| ≤

k

⇔ −

k

≤

E

≤

k

⇔

E

≥ −

k

and

E

≤

k

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Inequalities with Absolute Value

Let

E

be any real expression and let

k >

0. Then

|

E

|

< k

⇔ −

k < E < k

⇔

E >

−

k

and

E < k

|

E

| ≤

k

⇔ −

k

≤

E

≤

k

⇔

E

≥ −

k

and

E

≤

k

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Inequalities with Absolute Value

Let

E

be any real expression and let

k >

0. Then

|

E

|

< k

⇔ −

k < E < k

⇔

E >

−

k

and

E < k

|

E

| ≤

k

⇔ −

k

≤

E

≤

k

⇔

E

≥ −

k

and

E

≤

k

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Inequalities with Absolute Value

Let

E

be any real expression and let

k >

0. Then

|

E

|

< k

⇔ −

k < E < k

⇔

E >

−

k

and

E < k

|

E

| ≤

k

⇔ −

k

≤

E

≤

k

⇔

E

≥ −

k

and

E

≤

k

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Find the solution set of|x−1| ≤3 Solution:

−3≤ x−1 and x−1 ≤ 3

[−2,+∞) ∩ (−∞,4]

Hence, the solution set is [−2,4].

Another solution:

−3 ≤ x−1 ≤ 3

1−3 ≤ x ≤ 3 + 1

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Find the solution set of|x−1| ≤3 Solution:

−3≤ x−1 and x−1 ≤ 3

−2 ≤ x and x ≤ 4

[−2,+∞) ∩ (−∞,4]

Hence, the solution set is [−2,4].

Another solution:

−3 ≤ x−1 ≤ 3

1−3 ≤ x ≤ 3 + 1

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Find the solution set of|x−1| ≤3 Solution:

−3≤ x−1 and x−1 ≤ 3

−2 ≤ x and x ≤ 4

[−2,+∞) ∩ (−∞,4]

Another solution:

−3 ≤ x−1 ≤ 3

1−3 ≤ x ≤ 3 + 1

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Find the solution set of|x−1| ≤3 Solution:

−3≤ x−1 and x−1 ≤ 3

−2 ≤ x and x ≤ 4

[−2,+∞) ∩ (−∞,4]

Hence, the solution set is [−2,4].

Another solution:

−3 ≤ x−1 ≤ 3

1−3 ≤ x ≤ 3 + 1

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Find the solution set of|x−1| ≤3 Solution:

−3≤ x−1 and x−1 ≤ 3

−2 ≤ x and x ≤ 4

[−2,+∞) ∩ (−∞,4]

Hence, the solution set is [−2,4].

Another solution:

−3 ≤ x−1 ≤ 3

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Find the solution set of|x−1| ≤3 Solution:

−3≤ x−1 and x−1 ≤ 3

−2 ≤ x and x ≤ 4

[−2,+∞) ∩ (−∞,4]

Hence, the solution set is [−2,4].

Another solution:

−3 ≤ x−1 ≤ 3

1−3 ≤ x ≤ 3 + 1

Find the solution set of|x−1| ≤3 Solution:

−3≤ x−1 and x−1 ≤ 3

−2 ≤ x and x ≤ 4

[−2,+∞) ∩ (−∞,4]

Hence, the solution set is [−2,4].

Another solution:

−3 ≤ x−1 ≤ 3

1−3 ≤ x ≤ 3 + 1

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Find the solution set of|x−1| ≤3 Solution:

−3≤ x−1 and x−1 ≤ 3

−2 ≤ x and x ≤ 4

[−2,+∞) ∩ (−∞,4]

Hence, the solution set is[−2,4].

Another solution:

−3 ≤ x−1 ≤ 3

1−3 ≤ x ≤ 3 + 1

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example : Find all real values of xthat satisfy

x x−2

≤1

Solution:

−1 ≤ x

x−2 and

x

x−2 ≤ 1

0 ≤

x−2+ x−2 −x−2+ x−2 ≤ 0 0 ≤ 2x−2

x−2 and

2

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example : Find all real values of xthat satisfy

x x−2

≤1 Solution:

−1 ≤ x

x−2 and

x

x−2 ≤ 1

0 ≤ x

x−2+ x−2

x−2 −

x−2 x−2+

x

x−2 ≤ 0

0 ≤ 2x−2

x−2 and

2

Example : Find all real values of xthat satisfy x x−2

≤1 Solution:

−1 ≤ x

x−2 and

x

x−2 ≤ 1

0 ≤ x

x−2+ x−2

x−2 −

x−2 x−2+

x

x−2 ≤ 0 0 ≤ 2x−2

x−2 and

2

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example : Find all real values of xthat satisfy

x x−2

≤1

Solution (cont):

For 2x−2

x−2 ≥0 :

Table of Signs:

(−∞,1) (1,2) (2,∞) 2(x−1)

− + +

x−2

− − +

2(x−1) x−2

+ − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example : Find all real values of xthat satisfy

x x−2

≤1

Solution (cont):

For 2x−2

x−2 ≥0 :

Table of Signs:

(−∞,1) (1,2) (2,∞) 2(x−1)

x−2

− − +

2(x−1) x−2

+ − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example : Find all real values of xthat satisfy

x x−2

≤1

Solution (cont):

For 2x−2

x−2 ≥0 :

Table of Signs:

(−∞,1) (1,2) (2,∞)

2(x−1) −

+ +

x−2

− − +

2(x−1) x−2

+ − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example : Find all real values of xthat satisfy

x x−2

≤1

Solution (cont):

For 2x−2

x−2 ≥0 :

Table of Signs:

(−∞,1) (1,2) (2,∞)

2(x−1) − +

x−2

− − +

2(x−1) x−2

+ − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example : Find all real values of xthat satisfy

x x−2

≤1

Solution (cont):

For 2x−2

x−2 ≥0 :

Table of Signs:

(−∞,1) (1,2) (2,∞)

2(x−1) − + +

x−2

− − +

2(x−1) x−2

+ − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example : Find all real values of xthat satisfy

x x−2

≤1

Solution (cont):

For 2x−2

x−2 ≥0 :

Table of Signs:

(−∞,1) (1,2) (2,∞)

2(x−1) − + +

x−2 −

2(x−1) x−2

+ − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example : Find all real values of xthat satisfy

x x−2

≤1

Solution (cont):

For 2x−2

x−2 ≥0 :

Table of Signs:

(−∞,1) (1,2) (2,∞)

2(x−1) − + +

x−2 − −

+

2(x−1) x−2

+ − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example : Find all real values of xthat satisfy

x x−2

≤1

Solution (cont):

For 2x−2

x−2 ≥0 :

Table of Signs:

(−∞,1) (1,2) (2,∞)

2(x−1) − + +

x−2 − − +

2(x−1) x−2

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example : Find all real values of xthat satisfy

x x−2

≤1

Solution (cont):

For 2x−2

x−2 ≥0 :

Table of Signs:

(−∞,1) (1,2) (2,∞)

2(x−1) − + +

x−2 − − +

2(x−1)

x−2 +

− +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example : Find all real values of xthat satisfy

x x−2

≤1

Solution (cont):

For 2x−2

x−2 ≥0 :

Table of Signs:

(−∞,1) (1,2) (2,∞)

2(x−1) − + +

x−2 − − +

2(x−1)

x−2 + −

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example : Find all real values of xthat satisfy

x x−2

≤1

Solution (cont):

For 2x−2

x−2 ≥0 :

Table of Signs:

(−∞,1) (1,2) (2,∞)

2(x−1) − + +

x−2 − − +

2(x−1)

x−2 + − +

Example : Find all real values of xthat satisfy

x x−2

≤1

Solution (cont):

For 2x−2

x−2 ≥0 :

Table of Signs:

(−∞,1) (1,2) (2,∞)

2(x−1) − + +

x−2 − − +

2(x−1)

x−2 + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example : Find all real values of xthat satisfy

x x−2

≤1

Solution (cont):

For 2

x−2 ≤0,

note that the numerator is always positive.

So solution set isSSB = (−∞,2). Getting the intersection,

SSA∩SSB= ((−∞,1]∪(2,∞))∩(−∞,2)

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example : Find all real values of xthat satisfy

x x−2

≤1

Solution (cont):

For 2

x−2 ≤0, note that the numerator is always positive. So solution set is

B

Getting the intersection,

SSA∩SSB= ((−∞,1]∪(2,∞))∩(−∞,2)

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example : Find all real values of xthat satisfy

x x−2

≤1

Solution (cont):

For 2

x−2 ≤0, note that the numerator is always positive. So solution set isSSB = (−∞,2).

Getting the intersection,

SSA∩SSB= ((−∞,1]∪(2,∞))∩(−∞,2)

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example : Find all real values of xthat satisfy

x x−2

≤1

Solution (cont):

For 2

x−2 ≤0, note that the numerator is always positive. So solution set isSSB = (−∞,2).

Getting the intersection,

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example : Find all real values of xthat satisfy

x x−2

≤1

Solution (cont):

For 2

x−2 ≤0, note that the numerator is always positive. So solution set isSSB = (−∞,2).

Getting the intersection,

SSA∩SSB= ((−∞,1]∪(2,∞))∩(−∞,2)

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Inequalities with Absolute Value

Let

E

be any real expression and let

k >

0. Then

|

E

|

> k

⇔

E <

−

k

or

E > k

|

E

| ≥

k

⇔

E

≤ −

k

or

E

≥

k

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Inequalities with Absolute Value

Let

E

be any real expression and let

k >

0. Then

|

E

|

> k

⇔

E <

−

k

or

E > k

|

E

| ≥

k

⇔

E

≤ −

k

or

E

≥

k

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Inequalities with Absolute Value

Let

E

be any real expression and let

k >

0. Then

|

E

|

> k

⇔

E <

−

k

or

E > k

|

E

| ≥

k

⇔

E

≤ −

k

or

E

≥

k

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Inequalities with Absolute Value

Let

E

be any real expression and let

k >

0. Then

|

E

|

> k

⇔

E <

−

k

or

E > k

|

E

| ≥

k

⇔

E

≤ −

k

or

E

≥

k

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find the solution set of|2x−1| ≥4 Solution:

2x−1 ≤ −4

x ≤ −3_/ 2

or

2x−1 ≥ 4

x ≥ 5_/ 2

Hence, the solution set is (−∞,−3_/

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find the solution set of|2x−1| ≥4 Solution:

2x−1 ≤ −4

2x ≤ −4 + 1 x ≤ −3_/

2

or

2x−1 ≥ 4

2x ≥ 4 + 1 x ≥ 5_/

2

Hence, the solution set is (−∞,−3_/

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find the solution set of|2x−1| ≥4 Solution:

2x−1 ≤ −4 2x ≤ −4 + 1

x ≤ −3_/ 2

or

2x−1 ≥ 4

x ≥ 5_/ 2

Hence, the solution set is (−∞,−3_/

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find the solution set of|2x−1| ≥4 Solution:

2x−1 ≤ −4 2x ≤ −4 + 1

x ≤ −3_/ 2

or

2x−1 ≥ 4

2x ≥ 4 + 1 x ≥ 5_/

2

Hence, the solution set is (−∞,−3_/

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find the solution set of|2x−1| ≥4 Solution:

2x−1 ≤ −4 2x ≤ −4 + 1

x ≤ −3_/ 2

or

2x−1 ≥ 4 2x ≥ 4 + 1

x ≥ /2

Hence, the solution set is (−∞,−3_/

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find the solution set of|2x−1| ≥4 Solution:

2x−1 ≤ −4 2x ≤ −4 + 1

x ≤ −3_/ 2

or

2x−1 ≥ 4 2x ≥ 4 + 1

x ≥ 5_/ 2

Hence, the solution set is (−∞,−3_/

Example: Find the solution set of|2x−1| ≥4 Solution:

2x−1 ≤ −4 2x ≤ −4 + 1

x ≤ −3_/ 2

or

2x−1 ≥ 4 2x ≥ 4 + 1

x ≥ 5_/ 2

Hence, the solution set is (−∞,−3_/

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x that satisfy

x+ 1 x−1

≥2. Solution:

x+ 1

x−1 ≤ −2 or

x+ 1 x−1 ≥ 2

2

x−1 x−1

+ x+ 1

x−1 ≤ 0 −2

x−1 x−1

+ x+ 1

x−1 ≥ 0 3x−1

x−1 ≤ 0 or

−x+ 3

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x that satisfy

x+ 1 x−1

≥2. Solution:

x+ 1

x−1 ≤ −2 or

x+ 1 x−1 ≥ 2

2 x−1 x−1 +

x+ 1

x−1 ≤ 0 −2

x−1 x−1 +

x+ 1

x−1 ≥ 0 3x−1

x−1 ≤ 0 or

−x+ 3

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x that satisfy

x+ 1 x−1

≥2. Solution:

x+ 1

x−1 ≤ −2 or

x+ 1 x−1 ≥ 2

2

x−1 x−1

+ x+ 1

x−1 ≤ 0

−2

x−1 x−1

+ x+ 1

x−1 ≥ 0 3x−1

x−1 ≤ 0 or

−x+ 3

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x that satisfy

x+ 1 x−1

≥2. Solution:

x+ 1

x−1 ≤ −2 or

x+ 1 x−1 ≥ 2

2

x−1 x−1

+ x+ 1

x−1 ≤ 0

−2 x−1 x−1 +

x+ 1

x−1 ≥ 0

3x−1

x−1 ≤ 0

or −x+ 3

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x that satisfy

x+ 1 x−1

≥2. Solution:

x+ 1

x−1 ≤ −2 or

x+ 1 x−1 ≥ 2

2

x−1 x−1

+ x+ 1

x−1 ≤ 0 −2

x−1 x−1

+ x+ 1

x−1 ≥ 0 3x−1

x−1 ≤ 0

or −x+ 3

Example: Find all real values of x that satisfy

x+ 1 x−1

≥2. Solution:

x+ 1

x−1 ≤ −2 or

x+ 1 x−1 ≥ 2

2

x−1 x−1

+ x+ 1

x−1 ≤ 0 −2

x−1 x−1

+ x+ 1

x−1 ≥ 0 3x−1

x−1 ≤ 0 or

−x+ 3

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x that satisfy

x+ 1 x−1

≥2. Solution(cont):

For 3x−1

x−1 ≤0 :

Table of Signs:

(−∞,1_/

3) (1/3, 1) (1,∞)

3x−1

− + +

x−1

− − +

3x−1 x−1

+ − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x that satisfy

x+ 1 x−1

≥2.

Solution(cont):

For 3x−1

x−1 ≤0 :

Table of Signs:

(−∞,1_/

3) (1/3, 1) (1,∞)

3x−1 x−1

− − +

3x−1 x−1

+ − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x that satisfy

x+ 1 x−1

≥2. Solution(cont):

For 3x−1

x−1 ≤0 :

Table of Signs:

(−∞,1_/

3) (1/3, 1) (1,∞)

3x−1 −

+ +

x−1

− − +

3x−1 x−1

+ − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x that satisfy

x+ 1 x−1

≥2.

Solution(cont):

For 3x−1

x−1 ≤0 :

Table of Signs:

(−∞,1_/

3) (1/3, 1) (1,∞)

3x−1 − +

x−1

− − +

3x−1 x−1

+ − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x that satisfy

x+ 1 x−1

≥2.

Solution(cont):

For 3x−1

x−1 ≤0 :

Table of Signs:

(−∞,1_/

3) (1/3, 1) (1,∞)

3x−1 − + +

x−1

− − +

3x−1 x−1

+ − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x that satisfy

x+ 1 x−1

≥2.

Solution(cont):

For 3x−1

x−1 ≤0 :

Table of Signs:

(−∞,1_/

3) (1/3, 1) (1,∞)

3x−1 − + +

x−1 −

3x−1 x−1

+ − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x that satisfy

x+ 1 x−1

≥2.

Solution(cont):

For 3x−1

x−1 ≤0 :

Table of Signs:

(−∞,1_/

3) (1/3, 1) (1,∞)

3x−1 − + +

x−1 − −

+

3x−1 x−1

+ − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x that satisfy

x+ 1 x−1

≥2.

Solution(cont):

For 3x−1

x−1 ≤0 :

Table of Signs:

(−∞,1_/

3) (1/3, 1) (1,∞)

3x−1 − + +

x−1 − − +

3x−1 x−1

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x that satisfy

x+ 1 x−1

≥2.

Solution(cont):

For 3x−1

x−1 ≤0 :

Table of Signs:

(−∞,1_/

3) (1/3, 1) (1,∞)

3x−1 − + +

x−1 − − +

3x−1

x−1 +

− +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x that satisfy

x+ 1 x−1

≥2.

Solution(cont):

For 3x−1

x−1 ≤0 :

Table of Signs:

(−∞,1_/

3) (1/3, 1) (1,∞)

3x−1 − + +

x−1 − − +

3x−1

x−1 + −

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x that satisfy

x+ 1 x−1

≥2.

Solution(cont):

For 3x−1

x−1 ≤0 :

Table of Signs:

(−∞,1_/

3) (1/3, 1) (1,∞)

3x−1 − + +

x−1 − − +

3x−1

x−1 + − +

Example: Find all real values of x that satisfy

x+ 1 x−1

≥2.

Solution(cont):

For 3x−1

x−1 ≤0 :

Table of Signs:

(−∞,1_/

3) (1/3, 1) (1,∞)

3x−1 − + +

x−1 − − +

3x−1

x−1 + − +

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x that satisfy

x+ 1 x−1

≥2.

Solution(cont):

For −x+ 3

x−1 ≥ 0 :

Table of Signs:

(−∞,1) (1,3) (3,∞)

−x+ 3

+ + −

x−1

− + +

3x−1 x−1

− + −

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x that satisfy

x+ 1 x−1

≥2.

Solution(cont):

For −x+ 3

x−1 ≥ 0 :

Table of Signs:

(−∞,1) (1,3) (3,∞)

−x+ 3 x−1

− + +

3x−1 x−1

− + −

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x that satisfy

x+ 1 x−1

≥2.

Solution(cont):

For −x+ 3

x−1 ≥ 0 :

Table of Signs:

(−∞,1) (1,3) (3,∞)

−x+ 3 +

+ −

x−1

− + +

3x−1 x−1

− + −

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x that satisfy

x+ 1 x−1

≥2.

Solution(cont):

For −x+ 3

x−1 ≥ 0 :

Table of Signs:

(−∞,1) (1,3) (3,∞)

−x+ 3 + +

x−1

− + +

3x−1 x−1

− + −

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x that satisfy

x+ 1 x−1

≥2.

Solution(cont):

For −x+ 3

x−1 ≥ 0 :

Table of Signs:

(−∞,1) (1,3) (3,∞)

−x+ 3 + + −

x−1

− + +

3x−1 x−1

− + −

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x that satisfy

x+ 1 x−1

≥2.

Solution(cont):

For −x+ 3

x−1 ≥ 0 :

Table of Signs:

(−∞,1) (1,3) (3,∞)

−x+ 3 + + −

x−1 −

3x−1 x−1

− + −

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x that satisfy

x+ 1 x−1

≥2.

Solution(cont):

For −x+ 3

x−1 ≥ 0 :

Table of Signs:

(−∞,1) (1,3) (3,∞)

−x+ 3 + + −

x−1 − +

+

3x−1 x−1

− + −

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x that satisfy

x+ 1 x−1

≥2.

Solution(cont):

For −x+ 3

x−1 ≥ 0 :

Table of Signs:

(−∞,1) (1,3) (3,∞)

−x+ 3 + + −

x−1 − + +

3x−1 x−1

Solution Set of Inequality Interval Notations Table of Signs Absolute Values

Example: Find all real values of x that satisfy

x+ 1 x−1