Absolute Value Inequalities

Algebra Level 3

Solve for x x in the inequality 2 < ∣ 3 − 2 x ∣ ≤ 3. 2<|3-2x|\leq 3.

0 ≤ x < 1 0 \leq x < 1 and 3 < x ≤ 3 3 < x \leq 3 0 ≤ x < 1 2 0 \leq x < \frac{1}{2} and 5 2 < x ≤ 3 \frac{5}{2} < x \leq 3

Stephen Liado by stephen liado , 21, Philippines

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1 solution

The inequality can be written as 2<|2(1.5-X)|=<3, then 2<2(|1.5-X|)=<3, and finally 1<|1.5-X|=<1.5.

We then have three cases: X<0 0=<X=<1.5 X>1.5

For X<0, the equation becomes 1.5 + X, which will always be greater than 1.5, and therefore not possible.

For 0=<X=<1.5, the inequality becomes 1<1.5-X=<1.5, which is also -0.5<-X=<0, and finally we are left with 0.5>X=>0.

For X>1.5, the inequality becomes 1<X-1.5=<1.5, which is also 2.5<X=<3.

And so the answers are 0=<X<1/2 and 5/2<X=<3.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Is there an easy ways or techniques , instead of cases?

stephen liado - 4 years, 7 months ago

Why does the solution contain the word 'and'. Doesn't that signify an intersection, and isn't this solution a union? There is no number that is a solution to both 0≤x≤0.5 and also 2.5<x≤3.

Rebecca P - 2 years, 7 months ago

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