# How do you solve the inequality abs(4x+7)< -3?

May 19, 2017

No solution , since $| 4 x + 7 | \ge 0$

#### Explanation:

$| 4 x + 7 | < - 3$ . No solution , since $| 4 x + 7 | \ge 0$

May 19, 2017

Is this question correct?
Assumption: The question is meant to read $| 4 x + 7 | < + 3$

With this assumption I get $\text{ } - \frac{5}{2} < x < - 1$

#### Explanation:

$\textcolor{b r o w n}{\text{By definition the outcome of "|4x+7|" can only be positive.}}$

$\textcolor{b r o w n}{\text{Thus it is not possible for the outcome to be less than negative 3}}$
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Assumption: The question is meant to read $| 4 x + 7 | < + 3$

For this to be true we have: $| \pm 3 | = + 3$

Case 1: $\text{ } 4 x + 7 = - 3$

$\text{ } 4 x = - 10$
$\text{ } x = - \frac{5}{2}$

Case 2: $\text{ } 4 x + 7 = + 3$

$\text{ } 4 x = - 4$
$\text{ } x = - 1$
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This very likely defines a range of possible values that are between but not including $x = - 1 \mathmr{and} x = - \frac{5}{2} \text{ " =>" } - \frac{5}{2} < x < - 1$

Lets test this by selecting values outside this domain

Set $x = - \frac{6}{2}$ giving $| 4 \left(- \frac{6}{2}\right) + 7 | \text{ " =" "|-12+7|=4 " not} < 3$

$\textcolor{w h i t e}{}$

Set $x = - \frac{9}{10}$ giving
$| 4 \left(- \frac{9}{10}\right) + 7 | \text{ "=" " |-18/5+7|" "=" "+3 2/5" not} < 3$
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color(red)("Thus we have: " -5/2 < x < -1