What is |z-1|+|z+1|<= 3 ?

The solution to |z-1|+|z+1|<= 3 is -3/2 <= z<= 3/2

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|z-1|+|z+1|<= 3

∣ z - 1 ∣ + ∣ z + 1 ∣ \leq 3

- \frac{3}{2} \leq z \leq \frac{3}{2}

Solution steps

∣ z - 1 ∣ + ∣ z + 1 ∣ \leq 3

Find positive and negative intervals

z < - 1, - 1 \leq z < 1, z \geq 1

Solve the inequality for each interval

- \frac{3}{2} \leq z < - 1 or - 1 \leq z < 1 or 1 \leq z \leq \frac{3}{2}

Merge Overlapping Intervals

- \frac{3}{2} \leq z \leq \frac{3}{2}

∣ 2 x + 3 ∣ > \frac{1}{2}

\frac{2 x - 3}{x - 1} > 3

\frac{( x + 1 )}{( x - 1 )} < 1

∣ x - 2 ∣ \leq ∣ x - 4 ∣

∣ 2 x - 1 ∣ > x + 1