Absolute value equations

$∣ x ∣$ measures how far $x$ is from 0. Distance is never negative. So $∣ x ∣ = 4$ has two answers, one on each side of zero.

Both 4 and

- 4

are 4 steps from zero.

y = ∣ x ∣

makes a V. Its point sits at the origin.

Work through it

Split into two cases. $x - 3 = 5$ or $x - 3 = - 5$

Solve each. $x = 8$ or $x = - 2$

Isolate first. Add 3. $2∣ x + 1∣ = 10$

Divide by 2. $∣ x + 1∣ = 5$

Split and solve. $x = 4$ or $x = - 6$

$∣ x ∣ = - 3$ gives $x = \pm 3$ .

Distance can't be negative. This equation has no solution.

The short version
Isolate the absolute value. Then split: what's inside equals the number, or its negative.

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