Question

solve for x.
4(x + 1)(x + 6) ≤ 0
write a compound inequality like 1 < x < 3 or like x < 1 or x > 3.

Explanation:

Step1: Analyze the coefficient

The coefficient of the quadratic part is \(4\), which is positive. The inequality is \(4(x + 1)(x + 6)\leq0\). Since \(4>0\), we can divide both sides by \(4\) without changing the inequality sign, getting \((x + 1)(x + 6)\leq0\).

Step2: Find critical points

Set each factor equal to zero: \(x + 1 = 0\) gives \(x=-1\), and \(x + 6 = 0\) gives \(x=-6\). These are the critical points that divide the number line into intervals.

Step3: Test intervals

• For \(x < -6\) (e.g., \(x=-7\)): \((-7 + 1)(-7 + 6)=(-6)(-1)=6>0\), so the inequality is not satisfied.
• For \(-6 < x < -1\) (e.g., \(x=-4\)): \((-4 + 1)(-4 + 6)=(-3)(2)=-6<0\), so the inequality is satisfied.
• For \(x > -1\) (e.g., \(x=0\)): \((0 + 1)(0 + 6)=(1)(6)=6>0\), so the inequality is not satisfied.

Step4: Include critical points

Since the inequality is \(\leq0\), we include the points where the expression equals zero, i.e., \(x=-6\) and \(x=-1\).

Answer:

\(-6\leq x\leq -1\)