Question

question 3
solve the following inequality and graph the solution:
(x + 8)(x + 8) ≤ 0
choose test values and indicate whether the inequality is true or false in each region.
test value inequality
empty boxes for test - value input dropdowns for inequality result input
write the solution as a compound inequality
empty box for compound - inequality input
and draw the solution
number line from 13 to - 2 with options to clear all and draw arrows, filled and empty circles
question help: video
calculator

Explanation:

Step1: Simplify the inequality

Since \((x + 8)(x + 8)=(x + 8)^2\), the inequality becomes \((x + 8)^2\leqslant0\).

Step2: Analyze the non - negativity of a square

We know that for any real number \(a\), \(a^2\geqslant0\). Here \(a=x + 8\). The only way for \((x + 8)^2\leqslant0\) to be true is when \((x + 8)^2 = 0\) because it cannot be negative.

Step3: Solve for \(x\)

Set \(x+8 = 0\), then \(x=-8\).

Step4: Choose test values

1. Choose \(x=-9\): \((-9 + 8)^2=(-1)^2 = 1>0\), so the inequality is false for \(x=-9\).
2. Choose \(x=-8\): \((-8 + 8)^2=0\), so the inequality is true for \(x=-8\).
3. Choose \(x=-7\): \((-7 + 8)^2=1^2 = 1>0\), so the inequality is false for \(x=-7\).

Answer:

Compound inequality: \(x=-8\)
On the number - line, we draw a filled - in circle at \(x = - 8\) (because \(x=-8\) is included in the solution set and there are no other values in the solution set).