How do you solve abs(7 + 2x) = 9?

May 21, 2018

$x = 1 , - 8$.

Explanation:

As The Absolute Value of the Expression is 9, we will have to solve the equation twice, once for positive and once for negative.

As. $| a | = a = | - a |$.

So, Case 1 (Taking Positive):

$\textcolor{w h i t e}{\times x} \left(7 + 2 x\right) = 9$

$\Rightarrow \cancel{7} + 2 x \cancel{- 7} = 9 - 7$ [Subtract $7$ from both sides]

$\Rightarrow 2 x = 2$

$\Rightarrow \frac{2 x}{2} = \frac{2}{2}$ [Dividing both sides by $2$]

$\Rightarrow x = 1$

Case 2 (Taking Negative) :

$\textcolor{w h i t e}{\times x} - \left(7 + 2 x\right) = 9$

$\Rightarrow - 7 - 2 x = 9$ [Distributive Property]

$\Rightarrow \cancel{- 7} - 2 x + \cancel{7} = 9 + 7$ [Add $7$ to both sides]

$\Rightarrow - 2 x = 16$

$\Rightarrow \frac{- 2 x}{-} 2 = \frac{16}{-} 2$ [Dividing both sides by $- 2$]

$\Rightarrow x = - 8$

So, $x$ has two values, $1 , - 8$.

Hope this helps.

May 21, 2018

$x = - 8 \text{ or } x = 1$

Explanation:

$\text{the expression inside the absolute value bars can be}$
$\text{positive or negative so there are 2 possible solutions}$

$\textcolor{m a \ge n t a}{\text{Positive expression}}$

$7 + 2 x = 9$

$\text{subtract 7 from both sides and divide by 2}$

$\Rightarrow 2 x = 9 - 7 = 2 \Rightarrow x = \frac{2}{2} = 1$

$\textcolor{m a \ge n t a}{\text{Negative expression}}$

$- \left(7 + 2 x\right) = 9$

$\Rightarrow - 7 - 2 x = 9$

$\text{add 7 to both sides and divide by } - 2$

$\Rightarrow - 2 x = 9 + 7 = 16 \Rightarrow x = \frac{16}{- 2} = - 8$

$\textcolor{b l u e}{\text{As a check}}$

Substitute these values into the left side of the equation and if equal to the right side then they are the solutions.

$x = 1 \to | 7 + 2 | = | 9 | = 9$

$x = - 8 \to | 7 - 16 | = | - 9 | = 9$

$\Rightarrow x = - 8 \text{ or "x=1" are the solutions}$