Absolute Value Inequalities

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Question 1 of 4

Solve for

x

$x$

| 2 x + 7 | \leq 5

$|2x+7|≤5$

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

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

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

$2 x + 7$ $2x+7$	$\leq$ $≤$	$5$ $5$
$2 x + 7$ $2x+7$ $- 7$ $-7$	$\leq$ $≤$	$5$ $5$ $- 7$ $-7$	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 $2$ $2$
$x$ $x$	$\leq$ $≤$	$- 1$ $-1$

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

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

$(2-3x)/4$ $times4$	$>$	$7$ $times4$	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 $2$

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

Absolute Value Inequalities

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Quiz summary

Information

1. Question

Hint

Well Done!

Representing Inequalities in the Number Line

2. Question

Hint

Nice Job!

Representing Inequalities in the Number Line

3. Question

Hint

Keep Going!

Representing Inequalities in the Number Line

4. Question

Hint

Correct!

Representing Inequalities in the Number Line

Quizzes

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

$6\|1/2 x+5\|$ $div6$	$<$	$6$ $div6$	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 $2$

$x$	$<$	$-8$

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

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

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

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

$x$	$>$	$-12$

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

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

$y$	$<$	$-3$