Solving Inequalities by Factoring

Flashcards covering quadratic inequalities and their solutions by factoring and using sign charts.

Quadratic Inequality

An inequality that can be solved by factoring the quadratic expression and analyzing the signs of the factors on a number line.

x² - 16 > 0

x² - 16 > 0 can be factored as (x - 4)(x + 4) > 0. Solutions are x < -4 or x > 4.

x² + 6x - 16 > 0

x² + 6x - 16 > 0 can be factored as (x + 8)(x - 2) > 0. Solutions are x < -8 or x > 2.

x² - 3x ≤ 10

x² - 3x ≤ 10 can be rearranged to x² - 3x - 10 ≤ 0, factored as (x - 5)(x + 2) ≤ 0. Solution is -2 ≤ x ≤ 5.

Sign Chart

A visual tool used to determine the intervals where a function is positive or negative by analyzing the signs of its factors.

What is a Quadratic Inequality?

An inequality involving a quadratic expression. Its solution involves finding intervals that satisfy the inequality.

How do you Solve a Quadratic Inequality?

Factor the quadratic expression and find critical points, then test intervals on a number line to determine where the inequality holds.

What are Critical Points in Quadratic Inequalities?

The values at which the quadratic expression equals zero; these points divide the number line into intervals for testing.

What is a Sign Chart?

A number line divided by critical points, used to test intervals for the sign of the quadratic expression.

How do you use Test Values in Solving Quadratic Inequalities?

Choose test values in each interval to determine whether the quadratic expression is positive or negative in that interval.

When do you Include or Exclude Critical Points in the Solution?

If the inequality is non-strict (≤ or ≥), include the critical points in the solution. If strict (< or >), exclude them.