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In Exercises 85–90, solve each inequality and graph the...

Key Concepts

Graphing on a Number Line

Graphing the solution set involves marking the critical points on a number line and shading the intervals that satisfy the inequality. Open or closed circle endpoints are used depending on whether the inequality is strict or includes equality, providing a visual representation of the solution.

Quadratic Inequalities

A quadratic inequality involves a quadratic expression, and the goal is to determine the values of the variable that make the inequality true. Solving such inequalities requires finding the zeros of the quadratic expression to divide the number line into intervals and then determining which intervals satisfy the inequality.

Finding Critical Points

Critical points, or roots, are the solutions to the corresponding quadratic equation obtained by setting the quadratic expression equal to zero. These points are essential because they determine the boundaries of the intervals where the quadratic expression might change its sign.

Interval Testing

After determining the critical points, the real number line is partitioned into intervals. A representative value (test point) is selected from each interval to evaluate the sign of the quadratic expression. This process helps identify the intervals that satisfy the inequality.

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