# Question #97535

Jan 15, 2018

$x = - 8 , - 4$

#### Explanation:

Get the absolute value by itself by subtracting $5$ from both sides and then dividing by $7$ to get $\left\mid x + 6 \right\mid = 2$. Since it is an absolute value, the answer $2$ could be reached with the contents of the absolute value being negative or positive, so there is two answers. These can be found with the equations $x + 6 = 2$ and $- \left(x + 6\right) = 2$. Simplify the first equation by subtracting $6$ from both sides to get $x = - 4$. Simplify the other equation by distributing the negative to get $- x - 6 = 2$, and add $6$ to both sides to get $- x = 8$. Divide $- 1$ from both sides to get $x = - 8$.

Jan 15, 2018

See a solution process below: $x = \left\{- 8 , - 4\right\}$

#### Explanation:

First, subtract $\textcolor{red}{5}$ from each side of the equation to isolate the absolute value term while keeping the equation balanced:

$5 - \textcolor{red}{5} + 7 \left\mid x + 6 \right\mid = 19 - \textcolor{red}{5}$

$0 + 7 \left\mid x + 6 \right\mid = 14$

$7 \left\mid x + 6 \right\mid = 14$

Now, divide each side of the equation by $\textcolor{red}{7}$ to isolate the absolute value function while keeping the equation balanced:

$\frac{7 \left\mid x + 6 \right\mid}{\textcolor{red}{7}} = \frac{14}{\textcolor{red}{7}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{7}}} \left\mid x + 6 \right\mid}{\cancel{\textcolor{red}{7}}} = 2$

$\left\mid x + 6 \right\mid = 2$

The absolute value function takes any term and transforms it to its non-negative form. Therefore, we must solve the term within the absolute value function for both its negative and positive equivalent.

Solution 1:

$x + 6 = - 2$

$x + 6 - \textcolor{red}{6} = - 2 - \textcolor{red}{6}$

$x + 0 = - 8$

$x = - 8$

Solution 2:

$x + 6 = 2$

$x + 6 - \textcolor{red}{6} = 2 - \textcolor{red}{6}$

$x + 0 = - 4$

$x = - 4$

The Solution Is:

$x = \left\{- 8 , - 4\right\}$

Jan 15, 2018

$x = - 4 \mathmr{and} x = - 8$

Have a look at https://socratic.org/help/symbols. Particularly note the hash symbols. They both open and close maths formatting.

#### Explanation:

The bits inside the | | are always interpreted so that the result is positive. That is $| \text{something} | \ge 0$. So it is permitted to take on the value of zero if it is appropriate. Not so in this case.

Given: $5 + 7 | x + 6 | = 19$

Note that $| x + 6 |$ is called: absolute $x + 6$

Treat the $| \text{something} |$ as being a special sort of brackets.

Subtract 5 from both sides

$7 | x + 6 | = 19 - 5$

Divide both sides by 7

$| x + 6 | = \frac{19 - 5}{7}$

$| x + 6 | = 2$

So now we have the condition:

$| \pm 2 | = + 2$

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The next step is to ask the question; what is happening within the 'absolute' value?

For the condition of inside the absolute we have $\pm 2$

For the condition of $| x + 6 | = | + 2 | \implies x = - 4$

For the condition of $| x + 6 | = | - 2 | \implies x = - 8$
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$\textcolor{b l u e}{\text{General comment}}$

Solutions are provided by volunteers. Sometimes they are answered quickly. Sometimes not so. Occasionally not at all. This normally occures if there is something wrong with the question

Formatting tip
hash 5+7|x+6|=19 hash gives $5 + 7 | x + 6 | = 19$

hash 2x xx3 hash gives $2 x \times 3$ but it goes wrong if you type
hash 2xxx3 hash $2 \times x 3$ Watch for variable x being next to xx for multiply. Do not use * for multiply. It gives a dot and looks like a decimal point.