The absolute value is defined as
[tex]|x|=\begin{cases}x&\text{if }x\ge0\\-x&\text{if }x<0\end{cases}[/tex]
So for example, if x = 3, then |x| = |3| = 3, since 3 is positive. On the other hand, if x = -5, then |x| = |-5| = -(-5) = 5, since -5 is negative. The absolute value is always positive.
For the inequality |7 + 8x| > 5, you consider the two cases where the argument to the absolute value (the expression you find inside the bars) is either positive or negative.
• If 7 + 8x ≥ 0, then |7 + 8x| = 7 + 8x, so that
[tex]|7+8x|>5\implies 7+8x>5 \implies 8x>-2 \implies x>-\dfrac14[/tex]
• Otherwise, if 7 + 8x < 0, then |7 + 8x| = -(7 + 8x), so that
[tex]|7+8x|>5\implies-(7+8x)>5\implies7+8x<-5\implies8x<-12\implies x<-\dfrac32[/tex]
The solution to the inequality is the union of these two intervals.
