# How do you solve 2-3abs(x-1)=-4abs(x-1)+7?

Jul 16, 2015

The solutions are $6$, and $- 4$
There are two possibilities $x \ge 1 \mathmr{and} x < 1$

#### Explanation:

(1) $x \ge 1$ so $x - 1$ is non-negative
The equality turns into:
$2 - 3 \left(x - 1\right) = - 4 \left(x - 1\right) + 7 \to$
$2 - 3 x + 3 = - 4 x + 4 + 7 \to$
$- 3 x + 4 x = 4 + 7 - 2 - 3 \to$
$x = 6$
(and this is $\ge 1$) -- always check this!

(2) $x < 1$ so $x - 1$ is negative. The absolute bars will turn the sign around and the equality turns into:
$2 - 3 \left(1 - x\right) = - 4 \left(1 - x\right) + 7 \to$
$2 - 3 + 3 x = - 4 + 4 x + 7 \to$
$3 x - 4 x = - 4 + 7 - 2 + 3 \to$
$- x = 4 \to x = - 4$
(and this is $< 1$)

Jul 16, 2015

The solutions are $6$ and $- 4$. Here is an approach with different details:

#### Explanation:

$2 - 3 \left\mid x - 1 \right\mid = - 4 \left\mid x - 1 \right\mid + 7$

Observe that the expression inside the absolute value signs is the same in both absolute values. It is $x - 1$. This approach will first find out what $\left\mid x - 1 \right\mid$ is equal to, and then, find out what $x$ must be.

Until you get some experience with working with expressions as if they were variables, it will help to actually do a substitution.
Our first goal is to find $\left\mid x - 1 \right\mid$. Since this is an absolute value, let's call it $a$ for the time being:

Let $a = \left\mid x - 1 \right\mid$.

Substituting, we have:

$2 - 3 a = - 4 a + 7$ Find $a$. (add $4 a$ and subtract $2$ on both sides)

$- 3 a + 4 a = 7 - 2$ (simplify)

$a = 5$

Good. We've finished step 1. Now we still need to find $x$. We'll use $a = \left\mid x - 1 \right\mid$ to write:

$\left\mid x - 1 \right\mid = 5$

The two numbers whose absolute value is $5$ are $- 5$ and $5$,, so

$x - 1 = - 5$ $\textcolor{w h i t e}{\text{xx}}$ or $\textcolor{w h i t e}{\text{xx}}$ $x - 1 = 5$ $\textcolor{w h i t e}{\text{xx}}$ so

$x = - 4$ $\textcolor{w h i t e}{\text{xx}}$ or $\textcolor{w h i t e}{\text{xx}}$ $x = 6$

Less details looks like this:

$2 - 3 \left\mid x - 1 \right\mid = - 4 \left\mid x - 1 \right\mid + 7$

$- 3 \left\mid x - 1 \right\mid + 4 \left\mid x - 1 \right\mid = 7 - 2$

$\left\mid x - 1 \right\mid = 5$

$x - 1 = - 5$ or $x - 1 = 5$

$x = - 4 \text{ or } 6$

Solutions: $- 4$, and $6$