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Algebra II : Solving Absolute Value Equations

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Example Questions

Example Question #51 : Absolute Value

Solve the equation: $\left | 2x+3\right | = 78$

Possible Answers:

$\frac{75}{2},\frac{81}{2}$

$-\frac{81}{2},\frac{75}{2}$

$-\frac{39}{2},\frac{75}{2}$

$-\frac{81}{2},-\frac{75}{2},\frac{75}{2},\frac{81}{2}$

$\textup{No solution.}$

Correct answer:

$-\frac{81}{2},\frac{75}{2}$

Explanation:

Break up the inequality and write out its positive and negative components.

2x+3=78

-(2x+3)=78

Solve the first equation. Subtract 3 from both sides.

2x+3-3=78-3

2x=75

Divide both sides by 2.

$\frac{2x}{2}=\frac{75}{2}$

The first solution is: $\frac{75}{2}$

Solve the second equation. Divide the equation by negative one, which will move the negative sign from left to right.

$\frac{-(2x+3)}{-1}=\frac{78}{-1}$

2x+3=-78

Subtract three from both sides.

2x+3-3=-78-3

2x=-81

Divide by 2 on both sides.

$\frac{2x}{2}=\frac{-81}{2}$

The second answer is: $x=-\frac{81}{2}$

The answers are: $-\frac{81}{2},\frac{75}{2}$

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Example Question #52 : Absolute Value

Solve the equation: $\left | 8x\right | -40= 35$

Possible Answers:

$\textup{There is no solution.}$

$\pm \frac{25}{4}$

$\pm \frac{75}{8}$

$\pm \frac{15}{8}$

$\pm \frac{5}{8}$

Correct answer:

$\pm \frac{75}{8}$

Explanation:

Add 40 on both sides of the equation.

$\left | 8x\right | -40+40= 35+40$

$\left | 8x\right | =75$

Split this absolute value into its positive and negative components.

8x=75

-(8x)=75

For the first equation, divide by eight to isolate the x-variable.

$\frac{8x}{8}=\frac{75}{8}$

The first answer is: $x=\frac{75}{8}$

For the second equation, divide by negative eight on both sides.

$\frac{-(8x)}{-8}=\frac{75}{-8}$

The second answer is: $x=-\frac{75}{8}$

The answer is: $\pm \frac{75}{8}$

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Example Question #53 : Absolute Value

Solve the equation: $\left |-9x \right | -18= 18$

Possible Answers:

$\pm\frac{1}{4}$

$\pm4$

$\textup{All real numbers.}$

$\textup{No solution.}$

Correct answer:

$\pm4$

Explanation:

Add 18 on both sides.

$\left |-9x \right | -18+(18)= 18+(18)$

Simplify both sides.

$\left |-9x \right | =36$

Separate the equation into its positive and negative components and eliminate the absolute value.

-9x=36

-(-9x)=36

Simplify the first equation by dividing both sides by negative nine.

$\frac{-9x}{-9}=\frac{36}{-9}$

x=-4

Simplify the second equation. Double negatives negate, turning the true sign to positive.

9x=36

Divide by nine on both sides.

$\frac{9x}{9}=\frac{36}{9}$

x=4

The answer is: $\pm4$

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Example Question #51 : Solving Absolute Value Equations

Solve for x: |2x-1| = 7

Possible Answers:

x = 4, -3

x=-4, -3

x=3, -4

x = 4, -4

Correct answer:

x = 4, -3

Explanation:

Whatever is inside the absolute value brackets could either be positive or negative.

Positive: 2x - 1 = 7

2x = 8

x = 4

Negative: 2x-1=-7

2x = -6

x=-3

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Example Question #4842 : Algebra Ii

Solve for x: -3|2x+5|=9x

Possible Answers:

x = 1

x=-1, 5

x=-1, -5

x=-1

Correct answer:

x=-1

Explanation:

First, divide both sides by -3:

|2x+5|=-3x

Whatever is inside the absolute value brackets could be positive or negative.

Positive: 2x+5 = -3x

5 = -5x

-1= x

Negative: 2x+5 = 3 x

5=x

Both of those answers may not work, however, so test both:

|2(-1)+5|=-3(-1)

|3| = 3

3=3

|2(5)+5|=-3(5)

|15| = -15

$15 \neq -15$

The only answer that works is -1.

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Example Question #52 : Solving Absolute Value Equations

Solve for x: |5x+3|-7=10

Possible Answers:

x=4, -2.8

x=4, -4

x=2.8, -4

x=-2.8, 2.8

Correct answer:

x=2.8, -4

Explanation:

First, add 7 to both sides:

|5x+3| = 17

Whatever is inside the absolute value brackets could either be positive or negative:

5x+3 = 17

5x = 14

x=2.8

5x+3 = -17

5x = -20

x=-4

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Example Question #4844 : Algebra Ii

Solve for x: 2|1-3x|+7 = 10

Possible Answers:

$x=-\frac{1}{6}$

$x=6, - \frac{5}{6}$

$x = - \frac{1}{6}, \frac{5}{6}$

$x = - \frac{1}{6}, -\frac{5}{6}$

$x= -\frac{3}{2} , -\frac{15}{2}$

Correct answer:

$x = - \frac{1}{6}, \frac{5}{6}$

Explanation:

First subtract 7 from both sides:

2|1-3x | = 3 divide both sides by 2

|1-3x | = 1.5

The expression inside the absolute value brackets could be positive or negative.

Positive: 1-3x = 1.5

$-3x = .5 = \frac{1}{}2$

$x=-\frac{1}{6}$

Negative: 1-3x = -1.5

$-3x = -2.5 = -\frac{5}{2}$

$x=\frac{5}{6}$

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Example Question #4845 : Algebra Ii

Solve for x: |3x+5| = 2x - 3

Possible Answers:

x=-0.4

No solution

x=-8, -0.4

x=8

x=-8

Correct answer:

No solution

Explanation:

The expression inside the absolute value brackets could be either positive or negative.

Positive:

3x+5 = 2x -3 subtract 5 from both sides

3x = 2x-8 subtract 2x from both sides

x=-8

Negative:

3x + 5 = -2x + 3 subtract 5 from both sides

3x = -2x -2 add 2x to both sides

5x = -2 divide both sides by 5

x=-0.4

Now see if either actually works:

|3(-8)+5| = 2(-8)-3

|-24+5| = -16 - 3

|-19| = -19

$19 \neq -19$

|3(-.4)+5| = 2(-.4)-3

|-1.2+5| = -.8 - 3

|3.8| = -3.8

$3.8 \neq -3.8$

Neither actually works.

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Example Question #59 : Absolute Value

Solve for x: |4x+1| -3 = 2x + 5

Possible Answers:

x=3.5, -1.5

x=3.5, -3.5

$x=-3.5, -\frac{7}{6}$

x=1.5, -3.5

Correct answer:

x=3.5, -1.5

Explanation:

First, add 3 to both sides:

|4x+1| = 2x + 8

Either the expression inside the brackets is positive or it is negative.

Positive: 4x+1 = 2x+8 subtract 1 from both sides

4x = 2x + 7 subtract 2x from both sides

2x = 7 divide both sides by 2

x=3.5

Negative: 4x+1 = -2x -8 subtract 1 from both sides

4x = -2x - 9 add 2x to both sides

6x = -9 divide both sides by 6

x=-1.5

Now check to be sure both actually work.

|4(3.5)+1| -3 = 2(3.5) + 5

|14 + 1 | - 3 = 7+5

15 - 3 = 12

12 = 12

|4(-1.5) + 1 | - 3 = 2(-1.5) + 5

|-6+1| -3 = -3 + 5

5-3 = -3 + 5

2= 2

Both work.

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Example Question #53 : Solving Absolute Value Equations

Solve for x: $\frac{1}{3} | x-4 | = 1 - 2x$

Possible Answers:

x=-0.2

x=0, -1

x=1, -0.2

x=-0.2, -1

Correct answer:

x=-0.2

Explanation:

First, multiply both sides by 3:

|x-4| = 3-6x

The expression inside the absolute value brackets could either be positive or negative.

Positive: x-4 = 3-6x add 6x to both sides

7x -4 = 3 add 4 to both sides

7x = 7 divide both sides by 7

x= 1

Negative: x-4 = 6x - 3 add 4 to both sides

x = 6x + 1 subtract 6x from both sides

-5 x = 1 divide both sides by -5

x=-0.2

Now test to see if both answers actually work.

$\frac{1}{3} |1-4 | = 1 - 2 (1)$

$\frac{1}{3} |-3| = 1-2$

$\frac{1}{3}\cdot 3 = -1$

$1 \neq -1$ so 1 does not work.

$\frac{1}{3} |-0.2 -4 | = 1-2(-0.2)$

$\frac{1}{3} |-4.2| = 1 + .4$

1.4 = 1.4

The answer is -0.2

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Algebra II : Solving Absolute Value Equations

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #51 : Absolute Value

Example Question #52 : Absolute Value

Example Question #53 : Absolute Value

Example Question #51 : Solving Absolute Value Equations

Example Question #4842 : Algebra Ii

Example Question #52 : Solving Absolute Value Equations

Example Question #4844 : Algebra Ii

Example Question #4845 : Algebra Ii

Example Question #59 : Absolute Value

Example Question #53 : Solving Absolute Value Equations

All Algebra II Resources

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