The variables of these equations are displayed in the second column.
The constant that is multiplied by the variable (appears in front of the variable) is called the coefficient of the variable. The last column in the above table displays the coefficients of the variables.
The expression in the last example, , is called a rational expression because it is the division of two expressions, 2x - 1 and 7x - 3. We can easily simplify it as follows:
= 2 2x - 4 = 2(7x - 3) 2x - 4 = 14x - 6 12x - 2 = 0. So the coefficient of x is 12.
Practice Exercises:
1. (Easy)
Write 3 examples of rational expression.
2. (Easy)
What is the variable and the coefficient in the following expressions?
a. -4d - 2 = 3 b. 2a3 - 8 = 1 c.
d. = 2
Answers:
1. (s4 -3s3 + 1)/(s2 + 2s);
2. a. d and -4; b. a and 2; c. x and 19; d. x and 6
How to Solve a One-Variable Simple Equation
Follow the steps below to solve a one-variable equation:
1. Perform all the operations to remove all the parentheses and rational expressions, if any. Order of the operations are explained for different expressions in chapter 5. If necessary, study those sections.
2. Carry the terms with the unknown to the one side of the equal sign. Remember that the sign of the term changes, from + to - and from - to +, as you move it from one side of equal sign to the other.
3. Carry the other terms to the other side of the equal sign. Remember that the sign of the term changes, from + to - and from - to +, as you move it from one side of equal sign to the other.
4. Add or subtract all the like terms on both sides of the equal sign.
5. Solve for the unknown by dividing both sides of the equal sign by the coefficient of the unknown.
Examples:
1. (Easy)
4x = 0, then x = 0 2. (Easy)
If -3b + 1 = 3, then b = ? Solution:
-3b + 1 = 3 -3b = 3 - 1 -3b = 2 b = -2/3 3. (Easy)
If -3a - 4 = 7a + 1, then a = ? Solution:
-3a - 4 = 7a + 1 -3a - 7a = 1 + 4 -10a = 5 a = -5/10 = -1/2
Equation Variable Coefficient
4x = 0 x 4 (of x)
-3b + 1 = 3 b -3 (of b)
2c2 - 3 = 15 c 2 (of c2) y2 - 2y + 2 = -2(y - 3) y 1 (of y2)
= 2 x 12 (of x)
2x 4– 7x 3–
---2x 4–
( ) 7x 3⁄( – )
2x 4– 7x 3–
---3x 4– 7x 8– –3x --- = –4 20x 4–
7x 3–
---Private Tutor for SAT Math Success 2006 | Algebra 7 - 3 4. (Easy)
If -4(5 - u) = 3(2u + 3), then u = ? Solution:
-4(5 - u) = 3(2u + 3) -20 + 4u = 6u + 9 4u - 6u = 9 + 20 -2u = 29 u = -29/2 5. (Easy)
If , then x = ?
Solution:
= 5 2x + 3 = 5 - 5x 7x = 2 x = 2/7 6. (Medium)
If 2c2 -3 = 15, then x = ? Solution:
2c2 -3 = 15 2c2 = 18 c2 = 18/2 = 9 c = 3 or c = -3.
Note that square of both 3 and -3 is 9.
Sometimes, the equation looks complicated with higher powers and/or more than one variables. You can simplify these equations by working through them.
Therefore if a question is in the beginning of a section and looks complicated, don’t be intimidated. Instead, try to simplify the question.
Examples:
1. (Medium)
If 2(x2 - 3x + 3) + x(3 - 2x) = x + 1, then x = ?
Solution:
2(x2 - 3x + 3) + x(3 - 2x) = x + 1 2x2 - 6x + 6 + 3x - 2x2 = x + 1 -3x + 6 = x + 1 6 - 1 = 3x + x 4x = 5 x = 5/4
2. (Medium)
If 2a - b - c2 - 12 = -(c2 - b + 1) + 2(a - 4), then a = ?, b = ? and c = ?
Solution:
2a - b - c2 - 12 = -(c2 - b + 1) + 2(a - 4) 2a - b - c2 - 12 = -c2 + b - 1 + 2a - 8 -b - b = 12 - 1 - 8 -2b = 3 b = -3/2 a and c can be any real number.
3. (Medium)
If y2 - 2y + 2 = -2(y - 3), then x = ?
Solution:
y2 - 2y + 2 = -2(y - 3) y2 - 2y + 2 = -2y + 6 y2 = 4 y = 2 or y = -2
Note that the square of both 2 and -2 is 4.
4. (Medium)
If , then x-2 = ?
Solution:
4(1 - x) = x(2x - 4) 4 - 4x = 2x2 - 4x 2x2 = 4 x2 = 2 x-2 = 1/2
Practice Exercises:
1. (Easy)
3x - 5 = 8, then x = ? 2. (Easy)
-3x - 5 = -8, then x = ? 3. (Easy)
3(x - 1) + 2x = 4x + 7, then x - 1 = ?
4. (Easy)
, then x = ?
5. (Medium)
2 - 7(c - 3 - 2c) + 5c = -6(-2c - 5) - 8 - c, then c = ?
6. (Medium)
5n2 + 9n = 3(n2 - 5) + 2n2, then 1/n = ?
7. (Medium)
, then = ?
8. (Medium)
Mark your solutions on a number line for the questions above.
Answers:
1. 13/3; 2. 1; 3. 9; 4. -5/11; 5. -1; 6. -3/5; 7. 0 or 8.
2x 3+ 1 x– --- = 5 2x 3+
1 x–
---2x 4– 1 x– --- 4
-x
=
2x 4– 1 x– --- 4
-x
=
2x 3+ 10 x– --- 1
-5
=
x 4– 1 – +2x --- –2
---x
= x– 2
2 2 –
.
0 10-2 -1
-3 Q5
. .
1.
Q2 2 3 4.
Q15 6 7 8 9Q3.
Q6
Question numbers are written in bold.
. .
Q7 Q7
Q4
Inequalities
One variable inequalities are similar in solution to one variable equalities, except the following:
• The equality sign is replaced by the inequality sign.
• When finding the unknown, you find a range of values instead of just one value.
Examples:
1. (Easy)
If x - 1 > 0, then x > 1. The solution to the inequality is a range, rather then a specific value. You can show this range on a number line as shown below.
2. (Easy)
• Important: when you multiply or divide both sides of an inequality with a negative number, the inequality sign changes its direction.
Examples:
1. (Easy)
If 2x > 1, then -2x < -1. Both sides of the inequality is multiplied by -1, hence the inequality sign is changed from “>” to “<“.
2. (Easy)
If , then . Both sides of the inequality is multiplied by -1, hence the inequality sign is changed from “ ” to
“ .”
3. (Easy)
-5a < 3, then a > -3/5. In this example, to find the unknown, you must divide both sides of the
inequality by -5, hence you must change the sign from “<” to “>”.
4. (Easy)
If , then . Both sides
of the inequality is multiplied by -1, hence the inequality sign is changed from “ ” to
“ .”
Note that the direction of the sign has changed in the last step.
6. (Medium)
If , what is the range of c?
Solution:
or .
Note that the square of the numbers greater than 3 and smaller than -3 are both greater than 9. greater than or equal to 0 (never negative),
Practice Exercises:
1. (Easy)
If , what is the range of x? Mark your solution on a number line.
-1 0 -2
-3 1 2 3 4 5 6 7
Note that 1 is excluded.
-1 0 -2
-3 1 2 3 4 5 6 7 8
Note that zero is excluded.
2x 3+
Note that 2/7 is included.
2/7
Note that -2/3 is included.
-2/3
2c2–3≥15
2c2–3≥15 2c2≥18 c2≥9 c 3≥ c≤–3
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Note that -3 and 3 are included.
.
Note that 0 is included and 2 is excluded.
3x 5– ≥8
Private Tutor for SAT Math Success 2006 | Algebra 7 - 5 2. (Medium)
If , what is the range of x? Mark your solution on a number line.
3. (Medium)
If 3(x - 1) + 2x < 4x + 7, what is the range of x? Mark your solution on a number line.
4. (Medium)
If 2 - 7(c - 3 -2c) + 5c > -6(-2c - 5) - 8 - c, what is the range of c? Mark your solution on a number line.
5. (Medium)
If 5n2 + 9n < 3(n2 - 5) + 2n2, what is the range of n?
Mark your solution on a number line.
6. (Medium)
If -3a - 4 > 7a + 1, what is the range of a? Mark your solution on a number line.
7. (Medium)
If -4(5 - u) < 3(2u + 3), what is the range of u? Mark your solution on a number line.
8. (Medium)
If , what is the range of x?
Mark your solution on a number line.
9. (Medium)
If , what is the range of x-2? Mark your solution on a number line.
Answers:
1. ; 2. ; 3. x < 10; 4. c > -1; 5. n < -5/3;
6. a < -1/2; 7. u > -29/2; 8. ; 9.
3
– x 5– ≥–8
2x 4– 1 x– --- 4
-x
<
2x 4– 1 x– --- 4
-x
<
x 13 ---3
≥ x 1≤
2
– < <x 2 x–2 1 -2
>
10 -1 0
-2
-3 1 2 3 4 5 6 7 8 9
.
Question 1. Note that 13/3 is included.
13/3
.
10 -1 0
-2
-3 1
.
2 3 4 5 6 7 8 9Question 2. Note that 1 is included.
10 -1 0
-2 1 2 3 4 5 6 7 8 9
Question 3. Note that 10 is excluded.
11
-6 -3 -2 -1 0 1 2 3 4 5 6
-7 -5 -4
Question 4. Note that -1 is excluded.
-6 -3 -2 -1 0 1 2 3 4 5 6
-7 -5 -4
Question 5. Note that -5/3 is excluded.
-5/3
-1 0 -2
-3 1 2 3 4 5 6 7 8
Question 6. Note that -1/2 is excluded.
-1/2
-12 -2 -14-13
-15 -11 -10 -9 -8 -7 -6 -5 -4 -3 -1 0 1 2 Question 7. Note that -29/2 is excluded.
-17-16 -29/2
-6 -3 -2 -1 0 1 2 3 4 5 6
-7 -5 -4
2 2 –
Question 8. Note that 2 and – 2 are excluded.
10 -1 0
-2
-3 1 2 3 4 5 6 7 8 9
Question 9. Note that 1/2 is excluded.
1/2