# How do you solve | 1/2-3/4x | + 1=3 ?

Mar 2, 2016

$-$2 and 10/3.

#### Explanation:

The given equation is equivalent to the pair 1/2$-$3 x/4 = $\pm$2.

Mar 2, 2016

$x = - 2 \text{ or "x=10/3" }$Note that $\frac{10}{3} = 3.33 \overline{3}$

#### Explanation:

The very nature of this type of problem is that it usually has 2 solutions for $x$

We need to end up with the 'Absolute' part of the equation on its own and on the left of the equals sign and everything else on the other.

Given:$\text{ } \textcolor{b r o w n}{| \frac{1}{2} - \frac{3}{4} x | + 1 = 3}$

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$\textcolor{b l u e}{\text{Simplifying}}$

Subtract $\textcolor{b l u e}{1}$ from both sides

" "color(brown)(|1/2-3/4 x|+1color(blue)(-1)" "=" "3color(blue)(-1))

$\text{ "|1/2-3/4 x|+0" "=" } 2$

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$\textcolor{b l u e}{\text{Solving for } x}$

Think of this as:

$\text{ "|1/2-3/4 x|" "=" } | \pm 2 |$

color(brown)("Consider: "1/2-3/4 x" "=" "-2" "

Multiply by (-1 ) giving

$\text{ } - \frac{1}{2} + \frac{3}{4} x = + 2$

$\text{ } \frac{3}{4} x = 2 + \frac{1}{2}$

$\textcolor{b l u e}{\text{ " x" "=" "4/3xx5/2" " = " } \frac{10}{3}}$
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color(brown)("Consider: "1/2-3/4 x=+2

Multiply by (-1 ) giving

$\text{ } - \frac{1}{2} + \frac{3}{4} x = - 2$

$\text{ } \frac{3}{4} x = - 2 + \frac{1}{2}$

$\text{ } \textcolor{b l u e}{x = \frac{4}{3} \times \left(- \frac{3}{2}\right) = - 2}$

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" "color(magenta)(x = -2" or "x=10/3) 