Linear Inequalities in One Variable

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Inequalities

Linear Inequalities in

One Variable

Inequalities

When you need to use an inequality to solve a word problem, you may encounter one of the phrases below. Important Words Sample

Sentence Equivalent Translation

is more than is greater than must exceed

Inequalities

When you need to use an inequality to solve a word problem, you may encounter one of the phrases below. Important Words Sample

Sentence Equivalent Translation

cannot exceed is at most is at least

Inequalities

You try a few: 1. 14 is greater than a 2. b is less than or equal to 8 3. 6 is less than the product of f and 20 4. The sum of t and 9 is greater than or equal to 36 5. 7 more than w is less than or equal to 10 6. 19 decreased by p is greater than or equal to 2 7. Fewer than 12 items 8. No more than 50 students 9. At least 275 people attended the play 1. 14 > a 2. b ≤ 8 3. 6 < 20f 4. t + 9 ≥ 36 5. 7 + w ≤ 10 6. 19 p ≥ 2 7. n < 12 8. s < 50 9. p > 275 Answers

Writing Inequalities

Do you speak math?

Try to change the following expressions from English into math. Twice a number is at most six. Two plus a number is at least four. 2x ≤ 6 2 + x ≥ 4 Answer Answer

Inequalities

Three less than a number is less than three times that number. The sum of two consecutive numbers is at least thirteen. Three times a number plus one is at least ten. x 3 < 3x x + (x + 1) ≥ 13 3x + 1 > 10 Answer Answer Answer

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Inequalities

A solution to an inequality is NOT a single number. It will have more than one value. 1 0 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 This would be read as the solution set is all numbers greater than or equal to negative 5.

Solution Sets

Simple Inequalites

•Solving onestep inequalities is much like solving onestep equations, with one exception. •To solve an inequality, you need to isolate the variable using the properties of inequalities and inverse operations. •When you multiply or divide both sides of an inequality by a negative number, the direction of the inequality reverses.

Simple Inequalities

1 0 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 5 > w  4  4 9 > w + 4 w < 5 C. 9 > w + 4 Solve and graph.

Simple Inequalities

1 0 2 3 4 5 1 2 3 4 5 5 6 2 A 1 0 2 3 4 5 1 2 3 4 5 B 1 0 2 3 4 5 1 2 3 4 5 C 1 0 2 3 4 5 1 2 3 4 5 D 1 Solve the inequality and graph the solution. 2 2 2 5 6 5 6 5 6

Simple Inequalities Mult/Div

2 A B C 50 > 5q 10 > q 10 < q 10 > q D 10 < q

Simple Inequalities

Words Original Inequality Multiply/ Divide by a Negative # Result Multiplying or dividing by a negative number reverses the inequality symbol 3 > 1 Multiply by 2 6 < 2 4 < 12 Divide by 4 1 > 3

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Simple Inequalities

The direction of the inequality changes only if the number you are using to multiply or divide by is negative .

Helpful Hint

Simple Inequalities

3 1 0 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Solve and graph. 5y ≤ 25

Simple Inequalities

4 1 0 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Solve and graph. n 2 > 3

Simple Inequalities

An inequality stays the same when you: 1. Add, subtract, multiply or divide by the same positive number on both sides

2. Add or subtract the same negative number on both sides

An inequality changes direction when you: 1. Multiply or divide by the same negative number on both sides

MultiStep Inequaltities

Solving TwoStep

and MultipleStep

Inequalities

MultiStep Inequalities

3x 10 = 14 3x 10 ≤ 14 You can solve two step inequalities in the same way you solve equations. You can add any positive or negative number to both sides of the inequality. You can multiply or divide both sides of an equality by any positive number. 3x 10 ≤ 14 + 10 +10 3x < 24 3 3 x < 8 is solved in the same way as

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MultiStep Inequalities

REMEMBER! If you multiply or divide by a negative number, reverse the direction of the inequality symbol! 3x ≤ 24 3 3 x ≥ 8

MultiStep Inequalities

1. Solve this twostep equation. 5 5x = 0 5 + 5x = 0 5 5 5x = 5 5 5 x = 1 Step 2: Use multiplicative inverse Step 1: Use additive inverse

MultiStep Inequalities

2. Solve this twostep inequality. 26 < 3n + 1 1 1 25 < 3n 3 3 8 < n 1 3 Step 2: Use multiplicative inverse Step 1: Use additive inverse

MultiStep Inequalities

4p 9 ≥ 23 + 9 +9 Add 9 to both sides 4p ≥ 32 Divide both sides by 4 4 4 (sign stays the same) p ≥ 8 In interval notation, the solution is Graph the solution { p | p ≥ 8 } Solve: 1 0 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Move to reveal graph

MultiStep Inequalities

5 A B C Solve and graph the solution. 18 < 4(x + 2) 2.5 < x 2.5 > x 2.5 < x D 2.5 > x

Multistep Inequalities

6 A B C Solve and graph the solution. 16 x > 7x 2 < x 2 > x 2 < x D 2 > x

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Simple Inequalities Mult/Div

7 A B C Solve and graph the solution. 36 > 3(x 5) 7 < x 7 > x 7 < x D 7 > x

TwoStep and MultiStep

Your town is having a fall carnival. Admission into the carnival is $3.00 and each game inside costs $0.25. Write an inequality that represents the possible number of games that can be played if you have $10.00. What is the maximum number of games that can be played? Hint : Ten dollars is the maximum amount of money that you have to spend at the carnival. What inequality symbol would be used?

TwoStep and MultiStep

ANSWER .25x + 3 ≤ 10 .25x + 3 ≤ 10  3 3 .25x ≤ 7 .25 .25 x ≤ 28 The maximum number of games that can be played is 28.

TwoStep Inequalities

You have $65.00 in birthday money and want to buy some CDs and a DVD. Suppose a DVD cost $15.00 and a CD cost $12.00. Write an inequality to find out how many CDs you can buy along with one DVD. Solve the inequality. Hint 1 The cost of 1 DVD and the unknown number of CDs must be less or equal to $65. Hint 2 How much does 1 CD cost? How would you express an unknown number of CDs?

TwoStep Inequalities

Pull down the shade to see the answer. 15 + 12x ≤ 65 15 + 12x ≤ 65 15 15 12x ≤ 50 12x ≤ 50 12 12 x ≤ 4.16 Can you buy 0.16 of a CD? You can buy 4 CDs and 1 DVD.

MultiStep Inequalities

8 Which graph represents the solution set for: A

B

C

D

Question from ADP Algebra I EndofCourse Practice Test 0 1 2 1 2 0 1 2 1 2 0 1 2 1 2 0 1 2 1 2 1 2 5 2 3 6x

<

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Multistep Inequalities

Using the Distributive Property

Solve Express the solution in interval notation and a number line graph. Use the distributive property Combine like terms Add x to both sides Add 10 to both sides Divide both sides by 2. Here you must reverse the inequality symbol from ≥ to ≤. The solution in interval notation is 1 0 2345678910 1 2 3 4 5 6 7 8 9 10 The graph of the solution is

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Solving a Linear Inequality with Fractions The solution in interval notation is Use the distributive property Multiply the terms on each side by 6, the least common denominator. Combine like terms Add 3r to both sides Subtract 9 from both sides Multiply both sides by 1. Remember to reverse the inequality sign.

MultiStep Inequalities

9

A

B

C

D

MultiStep Inequalities

10

A

B

C

D

3 Part Inequalities

ThreePart Inequalities We use threepart inequalities when the variable expression is between two numbers. Example Means x + 2 is between 3 and 8, exclusive. The inequality is written so that the symbols point in the same direction and the lesser number is on the left, the same order as on the number line.

3 Part Inequalities

Solving ThreePart Inequalities

Use the same steps as solving simple inequalities, but do the same thing to all three parts.

Add 1 to all three parts. Divide all three parts by 3. The inequality signs reverse because we divided by a negative number. Rewrite the inequality if necessary so that the lowest value is on the left, matching the order on the number line. The solution in interval notation looks like

Linear Inequalities in One Variable

Inequalities