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Algebra II : Solving Absolute Value Equations

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Example Question #61 : Solving Absolute Value Equations

Solve the absolute value: $\left | 3x-8\right | <6$

Possible Answers:

$(-\infty, \frac{2}{3})\bigcup ( \frac{2}{3}, \frac{14}{3})\bigcup (\frac{14}{3},\infty)$

$(-\infty, \infty)$

$(-\infty, \frac{2}{3})\bigcup (\frac{14}{3},\infty)$

$[\frac{2}{3}, \frac{14}{3}]$

$(\frac{2}{3}, \frac{14}{3})$

Correct answer:

$(\frac{2}{3}, \frac{14}{3})$

Explanation:

In order to solve this equation, we will need to eliminate the absolute value signs and rewrite the left term in its positive and negative components.

3x-8<6

-(3x-8)<6

Solve the first equation. Add eight on both sides.

3x-8+8<6+8

3x<14

Divide by three on both sides.

The first solution is: $x<\frac{14}{3}$

Divide the second equation by negative one. Dividing a negative value will require switching the direction of the sign.

$\frac{-(3x-8)}{-1}<\frac{6}{-1}$

3x-8>-6

Add eight on both sides.

3x-8+8>-6+8

3x>2

Divide by three on both sides.

$\frac{3x}{3}>\frac{2}{3}$

The second solution is: $x>\frac{2}{3}$

The answer is: $(\frac{2}{3}, \frac{14}{3})$

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Example Question #62 : Solving Absolute Value Equations

$\left | 4+x \right |> 9$

Possible Answers:

5> x> 13

-13> x> 13

-13> x> 5

0> x> 5

-5> x> 5

Correct answer:

-13> x> 5

Explanation:

Recall that when dealing with absolute value, you have to write the inequality two different ways to solve it:

1) 4+x> 9

2) 4+x< -9

Now, solve each one
1) x> 5

2) x< -13

Put those answers together to get your final answer:

-13> x> 5

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Example Question #63 : Solving Absolute Value Equations

Solve: $3\left | x-1\right |+7 =6$

Possible Answers:

$-\frac{2}{3}, \frac{4}{3}$

$\frac{3}{4}, 2$

$-\frac{3}{4}, \frac{2}{3}$

$\textup{No solution.}$

$-\frac{2}{3}, \frac{2}{3}$

Correct answer:

$\textup{No solution.}$

Explanation:

Start by isolating the absolute value.

Subtract seven on both sides.

$3\left | x-1\right |+7 -7=6-7$

$3\left | x-1\right |=-1$

Divide by three on both sides.

$\frac{3\left | x-1\right |}{3}=\frac{-1}{3}$

$\left | x-1\right |=-\frac{1}{3}$

There is no value of in an absolute value that will give a negative value since the absolute value will convert all numbers to positive.

The answer is: $\textup{No solution.}$

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Example Question #64 : Solving Absolute Value Equations

Solve for x: |3-x| = 19

Possible Answers:

x= 21, 16

x=-21, 21

x=-16, 16

x = -16, 22

Correct answer:

x = -16, 22

Explanation:

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

Positive: 3-x = 19

-x = 16

x = -16

Negative: 3-x = -19

-x = -22

x= 22

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Example Question #65 : Solving Absolute Value Equations

Solve: $\left | 8-2x\right |= 3x-2$

Possible Answers:

-6, -2

$\textup{There is no solution.}$

-2,6

-6, 2

Correct answer:

Explanation:

Separate the absolute value and solve both the positive and negative components of the absolute value.

8-2x = 3x-2

-(8-2x) = 3x-2

Solve the first equation. Add on both sides.

8-2x +2x= 3x-2+2x

8= 5x-2

Add two on both sides.

10=5x

Divide by five on both sides.

$\frac{10}{5}=\frac{5x}{5}$

One of the solutions is x=2 after substitution is valid.

Evaluate the second equation. Distribute the negative sign in the binomial.

-8+2x= 3x-2

Subtract on both sides.

-8+2x-2x= 3x-2-2x

-8 = x-2

Add two on both sides.

-8+2 = x-2+2

x=-6

If we substitute this value back to the original equation, the equation becomes invalid.

The answer is:

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Example Question #66 : Solving Absolute Value Equations

Solve the absolute value equation: $2\left | 3x-3\right | =16$

Possible Answers:

$\frac{5}{3}, \frac{11}{3}$

$-\frac{11}{3},-\frac{5}{3}$

$-\frac{5}{3}, \frac{11}{3}$

$-\frac{11}{3},\frac{5}{3}$

$\textup{There is no solution.}$

Correct answer:

$-\frac{5}{3}, \frac{11}{3}$

Explanation:

Divide by two on both sides to isolate the absolute value.

$\frac{2\left | 3x-3\right | }{2}=\frac{16}{2}$

The equation becomes:

$\left | 3x-3\right | = 8$

Split this equation into its positive and negative components.

3x-3 = 8

-(3x-3)=8

Solve the first equation. Add three on both sides.

3x-3+3 = 8+3

3x=11

Divide by three on both sides.

$\frac{3x}{3}=\frac{11}{3}$

The first solution is: $\frac{11}{3}$

Evaluate the second equation. Divide by negative one on both sides.

$\frac{-(3x-3)}{-1}=\frac{8}{-1}$

3x-3=-8

Add three on both sides.

3x-3+3=-8+3

3x=-5

Divide by three on both sides.

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

The second solution is: $-\frac{5}{3}$

The answers are: $-\frac{5}{3}, \frac{11}{3}$

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Example Question #67 : Solving Absolute Value Equations

Evaluate: $-3\left | 9x+2\right | = 33$

Possible Answers:

$-\frac{13}{9},-1$

$-\frac{13}{9},1$

$\textup{No solution.}$

$\frac{13}{9},1$

$\frac{13}{9},-1$

Correct answer:

$\textup{No solution.}$

Explanation:

Isolate the absolute value by dividing both sides by negative three.

$\frac{-3\left | 9x+2\right | }{-3}=\frac{ 33}{-3}$

The equation becomes:

$\left | 9x+2\right | = -11$

Recall that an absolute value cannot be negative since the absolute value will change negative values to positive.

The answer is: $\textup{No solution.}$

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Example Question #68 : Solving Absolute Value Equations

Solve: $\left | -3x+2\right |+3x = 6$

Possible Answers:

$\frac{1}{3}, \frac{4}{3}$

$\frac{4}{3}$

$\textup{There is no solution.}$

$-\frac{1}{3}, \frac{4}{3}$

$-\frac{3}{4}$

Correct answer:

$\frac{4}{3}$

Explanation:

Subtract on both sides.

$\left | -3x+2\right |+3x -3x= 6-3x$

$\left | -3x+2\right | = -3x+6$

Evaluate the positive and negative components of the absolute value. The two equations are:

-3x+2 = -3x+6

-(-3x+2 )= -3x+6

Evaluate the first equation. Add on both sides.

-3x+2 +3x= -3x+6+3x

2=6

There is no solution for the first equation.

Evaluate the second equation. Simplify the left side by distributing negative one through the binomial.

3x-2 = -3x+6

Add on both sides.

3x-2 +3x= -3x+6+3x

The equation becomes:

6x-2 = 6

Add two on both sides.

6x-2 +2= 6+2

6x=8

Divide by six on both sides and reduce the fraction. Verify that the answer will satisfy the original equation.

$\frac{6x}{6}=\frac{8}{6} = \frac{4}{3}$

The answer is: $\frac{4}{3}$

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Example Question #69 : Solving Absolute Value Equations

Solve the absolute value equation: $-2\left |\frac{1}{3}x-7 \right |-6 = -5$

Possible Answers:

$\frac{39}{2}$

$\textup{No solution.}$

$-\frac{45}{2}$

$\pm\frac{39}{2}$

$-\frac{45}{2},\frac{39}{2}$

Correct answer:

$\textup{No solution.}$

Explanation:

Evaluate by adding six on both sides.

$-2\left |\frac{1}{3}x-7 \right |-6+6 = -5+6$

$-2\left |\frac{1}{3}x-7 \right |=1$

Divide by negative two on both sides.

$\frac{-2\left |\frac{1}{3}x-7 \right |}{-2}=\frac{1}{-2}$

The equation becomes: $\left |\frac{1}{3}x-7 \right | = -\frac{1}{2}$

Recall that absolute values will convert all negative values to positive. No matter what would evaluate to be, it will never become negative half.

The answer is: $\textup{No solution.}$

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Example Question #70 : Solving Absolute Value Equations

Solve: $8\left |x-3 \right | = 7$

Possible Answers:

$-\frac{17}{8}, \frac{31}{8}$

$- \frac{31}{8},\frac{17}{8}$

$- \frac{31}{8},-\frac{17}{8}$

$\frac{17}{8}, \frac{31}{8}$

$\textup{No solution.}$

Correct answer:

$\frac{17}{8}, \frac{31}{8}$

Explanation:

Divide both sides by eight.

$\frac{8\left |x-3 \right | }{8}= \frac{7}{8}$

$\left |x-3 \right | = \frac{7}{8}$

Split the absolute value into its positive and negative components.

$x-3 = \frac{7}{8}$

$-(x-3 )= \frac{7}{8}$

Solve the first equation. Multiply both sides by eight to eliminate the fraction.

$8(x-3) = \frac{7}{8}\cdot 8$

8(x-3) = 7

Use distribution to simplify.

8x-24 = 7

Add 24 on both sides, and then divide both sides by eight.

8x-24 +24= 7+24

8x=31

$\frac{8x}{8}=\frac{31}{8}$

The first solution is $\frac{31}{8}$ .

Solve the second equation. Divide by negative one on both sides.

$\frac{-(x-3 )}{-1}= \frac{\frac{7}{8}}{-1}$

The equation becomes:

$x-3 = -\frac{7}{8}$

Add three on both sides. This is the same as adding $\frac{24}{8}$ on both sides.

$x-3+(3) = -\frac{7}{8}+(\frac{24}{8})$

$x=\frac{17}{8}$

The answers are: $\frac{17}{8}, \frac{31}{8}$

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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 #61 : Solving Absolute Value Equations

Example Question #62 : Solving Absolute Value Equations

Example Question #63 : Solving Absolute Value Equations

Example Question #64 : Solving Absolute Value Equations

Example Question #65 : Solving Absolute Value Equations

Example Question #66 : Solving Absolute Value Equations

Example Question #67 : Solving Absolute Value Equations

Example Question #68 : Solving Absolute Value Equations

Example Question #69 : Solving Absolute Value Equations

Example Question #70 : Solving Absolute Value Equations

All Algebra II Resources

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