Absolute Value Inequalities

Topics

Try VividMath Premium to unlock full access

Start Free Trial

Time limit: 0

Quiz summary

0 of 4 questions completed

Questions:

Information

–

You have already completed the quiz before. Hence you can not start it again.

Quiz is loading...

You must sign in or sign up to start the quiz.

You have to finish following quiz, to start this quiz:

Lets see your results!

Points

Score

Time

Retry Quiz

Answered
Review

Question 1 of 4

1. Question

Solve for

x

$x$

| 2 x + 7 | \leq 5

$|2x+7|≤5$

1. $- 6 > x > - 1$ $-6>x> -1$
2. $- 6 < x < - 1$ $-6<x<-1$
3. $- 6 \geq x \geq - 1$ $-6≥x≥-1$
4. $- 6 \leq x \leq - 1$ $-6≤x≤-1$

Hint

Help Video

Correct

Well Done!

Incorrect

Need Text

Play

Current Time 0:00

Duration Time 0:00

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

Video Acceleration:

Off

00:00

Mute

Playback Rate

2x
1.5x
1.25x
1x
0.75x
0.5x

Subtitles

subtitles off

Captions

captions off
English

Chapters

Chapters

Representing Inequalities in the Number Line

Greater than (

>

$>$ )

Greater than or equal (

\geq

$≥$ )

Less than (

<

$<$ )

Less than or equal (

\leq

$≤$ )

First, since we are solving an absolute value equation, form a positive and negative equation and solve for

x

$x$ on both equations.

For the negative value, switch the inequality sign.

Positive:

$2 x + 7$ $2x+7$	$\leq$ $≤$	$5$ $5$
$2 x + 7$ $2x+7$ $- 7$ $-7$	$\leq$ $≤$	$5$ $5$ $- 7$ $-7$	Subtract $7$ $7$ from both sides
$2 x$ $2x$	$\leq$ $≤$	$- 2$ $-2$
$2 x$ $2x$ $\div 2$ $div2$	$\leq$ $≤$	$- 2$ $-2$ $\div 2$ $div2$	Divide both sides by $2$ $2$
$x$ $x$	$\leq$ $≤$	$- 1$ $-1$

Negative:

$2 x + 7$ $2x+7$	$\geq$ $≥$	$- 5$ $-5$
$2 x + 7$ $2x+7$ $- 7$ $-7$	$\geq$ $≥$	$- 5$ $-5$ $- 7$ $-7$	Subtract $7$ $7$ from both sides
$2 x$ $2x$	$\geq$ $≥$	$- 12$ $-12$
$2 x$ $2x$ $\div 2$ $div2$	$\geq$ $≥$	$- 12$ $-12$ $\div 2$ $div2$	Divide both sides by $2$ $2$
$x$ $x$	$\geq$ $≥$	$- 6$ $-6$

Next, plot the values of

x

$x$ on the number line and draw the inequality.

Since the inequality has a less than or equal to sign,

x

$x$ should be between the known values.

Finally, write the value of

x

$x$ with the variable in the middle and the known values on each side

Known values

$x$ $x$	$\leq$ $≤$	$- 1$ $-1$
$- 6$ $-6$	$\leq$ $≤$	$x$ $x$	Same as $x \geq - 6$ $x≥-6$

- 6

$-6$

\leq

$≤$

x

$x$

\leq

$≤$

- 1

$-1$

- 6 \leq x \leq - 1

$-6≤x≤-1$

Question 2 of 4

2. Question

Plot the inequality

∣ ∣ ∣ \frac{2 - 3 x}{4} ∣ ∣ ∣ > 7

$|(2-3x)/4|>7$

Hint

Help Video

Correct

Nice Job!

Incorrect

Need Text

Play

Current Time 0:00

Duration Time 0:00

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

Video Acceleration:

Off

00:00

Mute

Playback Rate

2x
1.5x
1.25x
1x
0.75x
0.5x

Subtitles

subtitles off

Captions

captions off
English

Chapters

Chapters

Representing Inequalities in the Number Line

Greater than (

>

$>$ )

Greater than or equal (

\geq

$≥$ )

Less than (

<

$<$ )

Less than or equal (

\leq

$≤$ )

First, since we are solving an absolute value equation, form a positive and negative equation and solve for

x

$x$ on both equations.

For the negative value, switch the inequality sign.

Positive:

$\frac{2 - 3 x}{4}$ $(2-3x)/4$	$>$ $>$	$7$ $7$

$\frac{2 - 3 x}{4}$ $(2-3x)/4$ $\times 4$ $times4$	$>$ $>$	$7$ $7$ $\times 4$ $times4$	Multiply both sides by $4$ $4$

$2 - 3 x$ $2-3x$	$>$ $>$	$28$ $28$
$2 - 3 x$ $2-3x$ $- 2$ $-2$	$>$ $>$	$28$ $28$ $- 2$ $-2$	Subtract $2$ $2$ from both sides
$- 3 x$ $-3x$	$>$ $>$	$26$ $26$

$- 3 x$ $-3x$ $\div (- 3)$ $div(-3)$	$>$ $>$	$26$ $26$ $\div (- 3)$ $div(-3)$	Divide both sides by $- 3$ $-3$

$x$ $x$	$<$ $<$	$- \frac{26}{3}$ $-(26)/3$	Dividing both sides by a negative value reverses the inequality

$x$ $x$	$<$ $<$	$- 8 \frac{2}{3}$ $-8 2/3$	Convert to a mixed number

Negative:

$\frac{2 - 3 x}{4}$ $(2-3x)/4$	$<$ $<$	$- 7$ $-7$

$\frac{2 - 3 x}{4}$ $(2-3x)/4$ $\times 4$ $times4$	$<$ $<$	$- 7$ $-7$ $\times 4$ $times4$	Multiply both sides by $4$ $4$

$2 - 3 x$ $2-3x$	$<$ $<$	$- 28$ $-28$
$2 - 3 x$ $2-3x$ $- 2$ $-2$	$<$ $<$	$- 28$ $-28$ $- 2$ $-2$	Subtract $2$ $2$ from both sides
$- 3 x$ $-3x$	$<$ $<$	$- 30$ $-30$
$- 3 x$ $-3x$ $\div (- 3)$ $div(-3)$	$>$ $>$	$- 30$ $-30$ $\div (- 3)$ $div(-3)$	Divide both sides by $- 3$ $-3$
$x$ $x$	$>$ $>$	$10$ $10$	Dividing both sides by a negative value reverses the inequality

Next, plot the values of

x

$x$ on the number line and draw the inequality.

Question 3 of 4

3. Question

Solve for

x

$x$

6 ∣ ∣ ∣ \frac{1}{2} x + 5 ∣ ∣ ∣ < 6

$6|1/2 x+5|<6$

1. $- 12 < x < - 8$ $-12<x<-8$
2. $- 12 > x > - 8$ $-12>x> -8$
3. $- 8 < x < 12$ $-8<x<12$
4. $12 > x > - 8$ $12>x> -8$

Hint

Help Video

Correct

Keep Going!

Incorrect

Need Text

Play

Current Time 0:00

Duration Time 0:00

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

Video Acceleration:

Off

00:00

Mute

Playback Rate

2x
1.5x
1.25x
1x
0.75x
0.5x

Subtitles

subtitles off

Captions

captions off
English

Chapters

Chapters

Representing Inequalities in the Number Line

Greater than (

>

$>$ )

Greater than or equal (

\geq

$≥$ )

Less than (

<

$<$ )

Less than or equal (

\leq

$≤$ )

First, convert the equation to its standard form

$6 ∣ ∣ ∣ \frac{1}{2} x + 5 ∣ ∣ ∣$ $6\|1/2 x+5\|$	$<$ $<$	$6$ $6$

$6 ∣ ∣ ∣ \frac{1}{2} x + 5 ∣ ∣ ∣$ $6\|1/2 x+5\|$ $\div 6$ $div6$	$<$ $<$	$6$ $6$ $\div 6$ $div6$	Divide both sides by $6$ $6$

$∣ ∣ ∣ \frac{1}{2} x + 5 ∣ ∣ ∣$ $\|1/2 x+5\|$	$<$ $<$	$1$ $1$

Next, since we are solving an absolute value equation, form a positive and negative equation and solve for

x

$x$ on both equations.

For the negative value, switch the inequality sign.

Positive:

$\frac{1}{2} x + 5$ $1/2 x+5$	$<$ $<$	$1$ $1$

$\frac{1}{2} x + 5$ $1/2 x+5$ $- 5$ $-5$	$<$ $<$	$1$ $1$ $- 5$ $-5$	Subtract $5$ $5$ from both sides

$\frac{1}{2} x$ $1/2 x$	$<$ $<$	$- 4$ $-4$

$\frac{1}{2} x$ $1/2 x$ $times2$	$<$	$-4$ $times2$	Multiply both sides by $2$

$x$	$<$	$-8$

Negative:

$1/2 x+5$	$>$	$-1$

$1/2 x+5$ $-5$	$>$	$-1$ $-5$	Subtract $5$ from both sides

$1/2 x$	$>$	$-6$

$1/2 x$ $times2$	$>$	$-6$ $times2$	Multiply both sides by $2$

$x$	$>$	$-12$

Next, plot the values of

$x$ on the number line and draw the inequality.

Since the inequality has a less than sign,

$x$ should be between the known values.

Finally, write the value of

$x$ with the variable in the middle and the known values on each side

Known values

$x$	$<$	$-8$
$-12$	$<$	$x$	Same as $x$ $>$ $-12$

$-12$

$<$

$x$

$<$

$-8$

$-12$

$<$

$x$

$<$

$-8$

Question 4 of 4

4. Question

Plot the inequality

$2|1/3 y-4|-3>7$

Hint

Help Video

Correct

Correct!

Incorrect

Need Text

Play

Current Time 0:00

Duration Time 0:00

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

Video Acceleration:

Off

00:00

Mute

Playback Rate

2x
1.5x
1.25x
1x
0.75x
0.5x

Subtitles

subtitles off

Captions

captions off
English

Chapters

Chapters

Representing Inequalities in the Number Line

Greater than (

$>$ )

Greater than or equal (

$≥$ )

Less than (

$<$ )

Less than or equal (

$≤$ )

First, convert the equation to its standard form

$2\|1/3 y-4\|-3$	$>$	$7$

$2\|1/3 y-4\|-3$ $+3$	$>$	$7$ $+3$	Add $3$ to both sides

$2\|1/3 y-4\|$	$>$	$10$

$2\|1/3 y-4\|$ $div2$	$>$	$10$ $div2$	Divide both sides by $2$

$\|1/3 y-4\|$	$>$	$5$

Next, since we are solving an absolute value equation, form a positive and negative equation and solve for

$x$ on both equations.

For the negative value, switch the inequality sign.

Positive:

$1/3 y-4$	$>$	$5$

$1/3 y-4$ $+4$	$>$	$5$ $+4$	Add $4$ to both sides

$1/3 y$	$>$	$9$

$1/3 y$ $times3$	$>$	$9$ $times3$	Multiply both sides by $3$

$y$	$>$	$27$

Negative:

$1/3 y-4$	$<$	$-5$

$1/3 y-4$ $+4$	$<$	$-5$ $+4$	Add $4$ to both sides

$1/3 y$	$<$	$-1$

$1/3 y$ $times3$	$<$	$-1$ $times3$	Multiply both sides by $3$

$y$	$<$	$-3$

Next, plot the values of

$x$ on the number line and draw the inequality.

Quizzes

One Step Inequalities 1
One Step Inequalities 2
Two Step Inequalities
Multi-Step Inequalities 1
Multi-Step Inequalities 2
Compound Inequalities 1
Compound Inequalities 2
Compound Inequalities 3
Inequality Word Problems 1
Inequality Word Problems 2
Absolute Value Inequalities
Graph Linear Inequalities 1
Graph Linear Inequalities 2