Question

Determine the solution to the inequality. one third times the absolute value of the quantity x plus 1 end quantity is greater than or equal to 3 x ≤ −8 or x ≥ 8 x ≤ −10 or x ≤ 9 x ≤ −2 or x ≥ 2 x ≤ −10 or x ≥ 8

Answer 1

x ≤ −10 or x ≥ 8

Explanation:

You want the solution to 1/3|x +1| ≥ 3.

Solution

An absolute value equation is effectively a piecewise-defined equation. The domains of the pieces are bounded by the value of x that makes the absolute value function zero.

x < -1

In this region, the absolute value argument is negative, so the equation becomes ...

-1/3(x +1) ≥ 3

x +1 ≤ -9 . . . . . . . multiply by -3

x ≤ -10 . . . . . . . . . subtract 1 (This is consistent with x < -1.)

x ≥ -1

In this region, the absolute value argument is unchanged, so the equation is ...

1/3(x +1) ≥ 3

x +1 ≥ 9 . . . . . . . multiply by 3

x ≥ 8 . . . . . . . . . subtract 1 (This is consistent with x ≥ -1.)

The solution is x ≤ -10 or x ≥ 8.