Solving absolute value inequalities is not too hard as long as one knows how to write the absolute value inequality into two inequalities.

Example 1:

Solve the following inequality for x:

$$ 4|5x-6|=3 $$

First, isolate the absolute value expression on one side and the rest of the equation on the other:

$$ |5x-6|=\frac{3}{4} $$

Two equations must be created:

Solving this is simple for the first equation, as all you have to do is keep $5x-6$ and $3$ constant and set them equal to each other. This is basically:

$$ 5x-6=\frac{3}{4}\rightarrow5x=\frac{27}{4}\space\rightarrow x=\frac{27}{20} $$

For the second equation, keep the expression inside the absolute value constant but change the sign of the other part and set them equal. This is done because the inside of the absolute value can be negative or positive and the result will come out positive. So, if the inside of the absolute value is -3/4 instead of 3/4, it will still become 3/4 when put in the absolute value.

$$ 5x-6=-\frac{3}{4}\rightarrow5x=\frac{21}{4}\rightarrow x=\frac{21}{20} $$

So, the solution is x= 27/20 and x= 21/20.

<aside> 💡 Remember that if the equation was set as $|5x-6|=-3/4$ instead of $|5x+6|=3/4$, then there would be no solution because an absolute value can never equal a negative number.

</aside>

To do this:

• Keep the terms the same, for example, x+2 and 3 in this case, and keep the same inequality sign as given in the question. For example, $\space |x+2|\leq 3$ becomes $x+2\leq3$.
• Then, make one of the terms negative (x+2 or 3), preferably the smaller term (3), and change the sign of the inequality to the opposite. For example, $x+2\leq3$ becomes $\space -(x+2)\geq3$ OR, as shown above, $x+2\geq-(3)$.

Solve the above two equations. The answers are $x\leq1$ AND $x\geq-5$.

$$ x\leq1\space\space\text{U}\space\space x\geq-5= -5\leq x\leq1 $$