# How do you solve  |x – 4| > |3x – 1|?

Sep 13, 2017

$x \in \mathbb{R}$

#### Explanation:

When there is absolute value, we have to take two options the first where the first is positive the next is negative.
$| a + b |$
$= a + b$
or $= - \left(a + b\right) = - a - b$
|x–4|>|3x–1|

1. $= x - 4 > | 3 x - 1 |$
again doing the same with the left hand side:
$x - 4 > | 3 x - 1 |$

1a. $= x - 4 > 3 x - 1$

Subtracting x from both sides of the equation to get the unknown's terms to one side of the equation:
$\cancel{x} \cancel{- x} - 4 > 3 x - 1 - x$
$- 4 > 2 x - 1$

Adding 1 to both sides of the equation to isolate the unknown's term on the left hand side:
$- 4 + 1 > 2 x \cancel{- 1} \cancel{+ 1}$
$- 3 > 2 x$

Dividing both sides of the equation by 2 to isolate x:
$- \frac{3}{2} > \frac{\cancel{2} x}{\cancel{2}}$

1a. color(red)(x<-3/2

and

1b. $x - 4 > - \left(3 x - 1\right)$
$= x - 4 > - 3 x + 1$
Doing same as before to isolate x:
$\cancel{x} - 4 \cancel{- x} > - 3 x + 1 - x$
$- 4 > - 4 x$
$\frac{\cancel{- 4}}{\cancel{- 4}} > \frac{\cancel{- 4} x}{\cancel{- 4}}$
$1 > x$
1b. color(red)(x<1

and

1. $= - \left(x - 4\right) > | 3 x - 1 |$
$= - x + 4 > | 3 x - 1 |$

2a.$= - x + 4 > 3 x - 1$

simplifying: (adding 1 and subtracting x from both sides of the equation and then dividing by 4):
$\cancel{- x} \cancel{+ x} + 4 + 1 > 3 x \cancel{- 1} + x \cancel{+ 1}$
$4 + 1 > 3 x + x$
$5 > 4 x$

$\frac{5}{4} > \frac{\cancel{4} x}{\cancel{4}}$
$\frac{5}{4} > x$
2a.color(red)(x<5/4

and

2b.$= - x + 4 > - 1 \left(3 x - 1\right)$
$= - x + 4 > - 3 x + 1$

Simplifying by adding 3x to, subtracting 4 from both sides and then dividing by 2:
$- x \cancel{+ 4} + 3 x \cancel{- 4} > \cancel{- 3 x} + 1 \cancel{+ 3 x} - 4$
$- x + 3 x > 1 - 4$
$2 x \succ 3$
$\frac{\cancel{2} x}{\cancel{2}} > - \frac{3}{2}$
2b. $\textcolor{red}{x > - \frac{3}{2}}$

Simplifying 1a , 2a ,and 2b :
$x < - \frac{3}{2}$
$x < 1$
$x < \frac{5}{4} \rightarrow x < 1.25$
$\rightarrow x < - \frac{3}{2} < 1 < \frac{5}{4}$
rarr color(blue)(x<5/4

and

From 2a:
color(blue)(x> -3/2

since x is smaller than $\frac{5}{4}$ and bigger than $- \frac{3}{2}$ then x defined for any number.