Solve Quadratic Inequalities Worksheets [PDF]: Algebra 2 Math

How Will This Worksheet on "Solve Quadratic Inequalities" Benefit Your Student's Learning?

Reinforces understanding of quadratic inequalities and their solutions.
Develops critical thinking and analytical skills through problem-solving.
Strengthens algebraic proficiency in manipulating quadratic expressions.
Enhances graphical interpretation of quadratic inequalities on number lines or coordinate planes.
Applies math concepts to real-world scenarios through inequality modeling.
Promotes logical reasoning in interpreting and evaluating quadratic inequality solutions.
Prepares students for assessments with quadratic inequality-solving components.
Encourages self-directed learning and builds confidence in problem-solving abilities.

Ensure the quadratic inequality is in the form $ ax^2 + bx + c \, (<, \leq, >, \geq) \, 0 $.
Solve the equation $ ax^2 + bx + c = 0 $ to find the critical points. These roots (`x`-values) divide the number line into intervals.
Choose a test point from each interval between and beyond the roots.
Substitute these test points into the quadratic expression $ ax^2 + bx + c $ to determine if the expression is positive or negative in that interval.
Based on the sign in each interval and the inequality sign ( $<, \leq, >, \geq$ ), determine which intervals satisfy the inequality.
Include or exclude the roots based on whether the inequality is strict ($<,>$) or non-strict ($\leq, \geq$).
Write the solution as an interval or a union of intervals.

Solved Example

Q. Solve for

x

\newline

(x - 1)(x + 6) \leq 0

\newline

Write a compound inequality like

1 < x < 3

or like

x < 1

x > 3

Solution:

Plot Zeros and Test Intervals: Plot the zeros on a number line and test intervals. $\newline$ Intervals: $(-\infty, -6$ ), $(-6, 1$ ), $(1, \infty)$ .
Test $x = -7$ : Test $x = -7$ in $(x - 1)(x + 6)$ . $\newline$ $(-7 - 1)(-7 + 6) = (-8)(-1) = 8$ , which is $> 0$ .
Test $x = 0$ : Test $x = 0$ in $(x - 1)(x + 6)$ . $(0 - 1)(0 + 6) = (-1)(6) = -6$ , which is $< 0$ .
Test $x = 2$ : Test $x = 2$ in $(x - 1)(x + 6)$ . $(2 - 1)(2 + 6) = (1)(8) = 8$ , which is $> 0$ .
Combine Intervals: Combine the intervals where the product is $\leq 0$ . The solution is $-6 \leq x \leq 1$ .