What is |2x-3|<|x-5| ?

The solution to |2x-3|<|x-5| is -2<x< 8/3

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|2x-3|<|x-5|

∣ 2 x - 3 ∣ < ∣ x - 5 ∣

- 2 < x < \frac{8}{3}

Interval Notation

(- 2, \frac{8}{3})

Decimal Notation

- 2 < x < 2.66666 \dots

Solution steps

∣ 2 x - 3 ∣ < ∣ x - 5 ∣

Find positive and negative intervals

x < \frac{3}{2}, \frac{3}{2} \leq x < 5, x \geq 5

Solve the inequality for each interval

- 2 < x < \frac{3}{2} or \frac{3}{2} \leq x < \frac{8}{3} or No Solution

Merge Overlapping Intervals

- 2 < x < \frac{8}{3}

\frac{( 3 - ∣ x ∣ )}{( ∣ x ∣ - 5 )} > 0

- 5 ∣ x - 3 ∣ - 5 < 15

∣ 4 x - 3 ∣ \leq 4 - 3 x

∣ x - 28 ∣ \leq 2

x^{2} - 2 x - 3 > x^{2} + 7 x - 13