Absolute Value Inequalities

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Question 1 of 4

1. Question
Solve for $x$

$|2x+7|≤5$
- 1. $-6≥x≥-1$
- 2. $-6≤x≤-1$
- 3. $-6<x<-1$
- 4. $-6>x> -1$
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Representing Inequalities in the Number Line

Greater than ( $>$ )

Greater than or equal ( $≥$ )

Less than ( $<$ )

Less than or equal ( $≤$ )

First, 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:

$2x+7$ $≤$ $5$

$2x+7$ $-7$ $≤$ $5$ $-7$ Subtract $7$ from both sides

$2x$ $≤$ $-2$

$2x$ $div2$ $≤$ $-2$ $div2$ Divide both sides by $2$

$x$ $≤$ $-1$

Negative:

$2x+7$ $≥$ $-5$

$2x+7$ $-7$ $≥$ $-5$ $-7$ Subtract $7$ from both sides

$2x$ $≥$ $-12$

$2x$ $div2$ $≥$ $-12$ $div2$ Divide both sides by $2$

$x$ $≥$ $-6$

Next, plot the values of $x$ on the number line and draw the inequality.

Since the inequality has a less than or equal to 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

$x$ $≤$ $-1$

$-6$ $≤$ $x$ Same as $x≥-6$

$-6$ $≤$ $x$ $≤$ $-1$

$-6≤x≤-1$

Question 2 of 4

Plot the inequality

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

Question 3 of 4

3. Question
Solve for $x$

$6|1/2 x+5|<6$
- 1. $-8<x<12$
- 2. $-12>x> -8$
- 3. $-12<x<-8$
- 4. $12>x> -8$
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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

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

$6|1/2 x+5|$ $div6$ $<$ $6$ $div6$ Divide both sides by $6$

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

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:

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

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

$1/2 x$ $<$ $-4$

$1/2 x$ $times2$ $<$ $-4$ $times2$ 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

Plot the inequality

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

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

$(2-3x)/4$ $times4$	$>$	$7$ $times4$	Multiply both sides by $4$

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

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

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

$x$	$<$	$-8 2/3$	Convert to a mixed number

$(2-3x)/4$	$<$	$-7$

$(2-3x)/4$ $times4$	$<$	$-7$ $times4$	Multiply both sides by $4$

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

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

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

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

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

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

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

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

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

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

$y$	$>$	$27$

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

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

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

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

$y$	$<$	$-3$

Absolute Value Inequalities

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Quiz summary

Information

1. Question

Hint

Well Done!

Representing Inequalities in the Number Line

2. Question

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Nice Job!

Representing Inequalities in the Number Line

3. Question

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Keep Going!

Representing Inequalities in the Number Line

4. Question

Hint

Correct!

Representing Inequalities in the Number Line

Quizzes

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

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

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

$6\|1/2 x+5\|$ $div6$	$<$	$6$ $div6$	Divide both sides by $6$

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

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

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

$1/2 x$	$<$	$-4$

$1/2 x$ $times2$	$<$	$-4$ $times2$	Multiply both sides by $2$

$x$	$<$	$-8$