Solve each quadratic inequality. Give the solution set in interval notation. See Example 1. (a) (x - 5)(x + 2) ≥ 0 (b) (x - 5)(x + 2) > 0 (c) (x - 5)(x + 2) ≤ 0 (d) (x - 5)(x + 2) < 0

Verified step by step guidance

1. Identify the critical points by setting each factor equal to zero: $x-5=0$ and $x+2=0$. Solve these to find $x=5$ and $x=-2$.
2. Plot these critical points on a number line, dividing the line into intervals: $(-\mathrm{\infty },-2)$, $(-2,5)$, and $(5,\mathrm{\infty })$.
3. Choose a test point from each interval and substitute it into the inequality to determine if the interval satisfies the inequality. For example, test $x=-3$ for $(-\mathrm{\infty },-2)$, $x=0$ for $(-2,5)$, and $x=6$ for $(5,\mathrm{\infty })$.
4. For each inequality: (a) $(x-5)(x+2)\ge 0$, (b) $(x-5)(x+2)>0$, (c) $(x-5)(x+2)\le 0$, and (d) $(x-5)(x+2)<0$, determine which intervals satisfy the inequality based on the sign of the product in each interval.
5. Express the solution set for each inequality in interval notation, considering whether the endpoints are included (for $\ge$ or $\le$) or not (for $>$ or $<$).

Key Concepts

Quadratic Inequalities

Quadratic inequalities are expressions that involve a quadratic polynomial set in relation to a value, typically zero. They can be represented in forms such as (ax^2 + bx + c) > 0 or (ax^2 + bx + c) ≤ 0. Solving these inequalities involves determining the values of x for which the inequality holds true, often requiring the identification of critical points where the expression equals zero.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses parentheses and brackets to indicate whether endpoints are included (closed intervals) or excluded (open intervals). For example, the interval (2, 5] includes all numbers greater than 2 and up to 5, including 5 but not 2. This notation is essential for expressing the solution sets of inequalities succinctly.

Test Points Method

The test points method is a strategy used to determine the solution set of inequalities. After finding the critical points where the quadratic expression equals zero, one can select test points from the intervals created by these points. By substituting these test points back into the inequality, one can ascertain whether the inequality holds true in those intervals, thus identifying the solution set.