What is |(-2x+9)(x-1)|<5|x-1| ?

The solution to |(-2x+9)(x-1)|<5|x-1| is 2<x<7

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|(-2x+9)(x-1)|<5|x-1|

∣ (- 2 x + 9) (x - 1) ∣ < 5 ∣ x - 1 ∣

2 < x < 7

Interval Notation

(2, 7)

Solution steps

∣ (- 2 x + 9) (x - 1) ∣ < 5 ∣ x - 1 ∣

Find positive and negative intervals

x < 1, 1 \leq x \leq \frac{9}{2}, x > \frac{9}{2}

Solve the inequality for each interval

No Solution or 2 < x \leq \frac{9}{2} or \frac{9}{2} < x < 7

Merge Overlapping Intervals

2 < x < 7

∣ x - 3 ∣ - ∣ x - 6 ∣ > 1

∣ 4 x - 1 ∣ + 4 > 0

\frac{x ^{2} - 4 ∣ x ∣ + 3}{x ( x + 4 )} => 0

∣ x - 3 ∣ + 2 ∣ x + 1 ∣ > 4

\frac{x + 1}{x} < \frac{5}{2}