calculus , maths

Question

|x + 6| ≥ 2

solve and enter answer in interval notation

Answer 1

Assuming x + 2 > 0

x + 2 ≥ 2

x ≥ 0

Assuming x + 2 < 0

-(x + 2) ≥ 2

-x - 2 ≥ 2

-x ≥ 4

x ≤ -4

Answer 2

Hello Hami,

Since the problem considers the absolute value of x + 6, we must separately consider the case when x + 6 ≥ 0 and the case when x + 6 < 0.

When x + 6 ≥ 0, |x + 6| = x + 6.

Then, the inequality |x + 6| ≥ 2 simplifies to x + 6 ≥ 2, which solves to x ≥ -4, under the original condition that x + 6 ≥ 0; ie, under the original condition that x ≥ -6. Next, combining x ≥ -4 AND x ≥ -6 means that, overall, x ≥ -4.

When x + 6 < 0, |x + 6| = -(x + 6).

Then, the inequality |x + 6| ≥ 2 simplifies to -(x + 6) ≥ 2, which expands to –x – 6 ≥ 2 and solves to -8 ≥ x, under the original condition that x + 6 < 0; ie, under the original condition that x < -6. Next, combining -8 ≥ x AND x < -6 means that, overall, -8 ≥ x.

Last, combining these two sets of solutions, we get the final solution x ≥ -4 OR -8 ≥ x. So, on a number line, shade all points at and to the left of -8, as well as all points at and to the right of -4.

Hope this helps!