Solve |ax + b| < k three ways (distance, algebra and graph), spot inequations with no solution, and solve inequations with x in the denominator by multiplying by its square.
Absolute value inequations
$|x - a|$ is the distance between $x$ and $a$ on the number line. That gives the fastest way to read these inequations.
When $x$ has a coefficient, use algebra instead: $|ax + b| < k$ is the same as the double inequation $-k < ax + b < k$, and $|ax + b| > k$ splits into $ax + b < -k$ or $ax + b > k$.
Where is |x − a| below the line y = b?
Orange: y = |x − a|. Blue dashed: y = b. The shaded strip is where the V is below the line, a band of width 2b centred on a.
Try this
Set $b = 0$. Why does the band disappear? What would $|x - a| \ge 0$ give?
Example 1●
Solve $|x + 1| > 2$.
$x$ is more than $2$ from $-1$.
|x + 1| = |x − (−1)| is the distance from −1.
$x < -3$ or $x > 1$
Example 2●●
Solve $|2x - 3| \le 5$.
$-5 \le 2x - 3 \le 5$
$-2 \le 2x \le 8$
Add 3 to all three parts.
$-1 \le x \le 4$
Divide all three parts by 2.
Inequations with x in the denominator
You cannot simply multiply by $x - 4$, because you do not know whether it is positive or negative, so you do not know whether to reverse the sign.
Example 3●●●
Solve $\dfrac{5}{x - 4} \ge 1$.
Note $x \ne 4$.
The denominator cannot be zero.
$5(x - 4) \ge (x - 4)^2$
Multiply both sides by (x − 4)², which is positive.
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