Which of the following represents the values of x that are...

Which of the following represents the values of $x$ that are solutions for the inequality $(x+7)(8-2 x) \geq 0 ?$
A. $x \leq-7$ or $x \geq 4$
B. $x \leq-4$ or $x \geq 7$
C. $\frac{1}{7} \leq x \leq \frac{1}{4}$
D. $-4 \leq x \leq 7$
E. $-7 \leq x \leq 4$

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Key Concepts

Polynomial Inequalities

Polynomial inequalities involve determining the set of values for which a polynomial expression is either positive, negative, or zero. Solving these inequalities usually requires finding where the expression equals zero and analyzing the behavior of the expression in the intervals defined by these critical points.

Critical Points

Critical points are values that make each factor of the polynomial equal to zero. These points partition the number line into distinct intervals, and assessing the sign of the polynomial in each interval leads to identifying where the inequality holds.

Sign Analysis

Sign analysis involves testing points from the intervals determined by the critical points to decide whether the expression is positive or negative in that interval. This process is crucial for understanding how the polynomial changes its sign across different ranges on the number line.

Interval Notation

Interval notation is a concise way to represent the range of values that satisfy an inequality. It expresses continuous segments of the number line using brackets and parentheses to indicate whether endpoints are included or excluded from the solution set.

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