# Solve Quadratic Inequalities Worksheet

## 6 problems

Solve quadratic inequalities involves finding the set of values for a variable that satisfy an inequality where the expression is quadratic (i.e., involves a variable raised to the power of 2). These inequalities are solved similarly to quadratic equations, where the solutions are intervals on the number line rather than specific points. The process typically involves factoring or using the quadratic formula to determine the intervals where the inequality holds true.

Algebra 2
Inequalities

## 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.

## How to Solve Quadratic Inequalities?

• 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 $x < 1$ or $x > 3$.
Solution:
1. Plot Zeros and Test Intervals: Plot the zeros on a number line and test intervals.$\newline$Intervals: $(-\infty, -6$), $(-6, 1$), $(1, \infty)$.
2. Test $x = -7$: Test $x = -7$ in $(x - 1)(x + 6)$.$\newline$$(-7 - 1)(-7 + 6) = (-8)(-1) = 8$, which is $> 0$.
3. Test $x = 0$: Test $x = 0$ in $(x - 1)(x + 6)$.$(0 - 1)(0 + 6) = (-1)(6) = -6$, which is $< 0$.
4. Test $x = 2$: Test $x = 2$ in $(x - 1)(x + 6)$.$(2 - 1)(2 + 6) = (1)(8) = 8$, which is $> 0$.
5. Combine Intervals: Combine the intervals where the product is $\leq 0$. The solution is $-6 \leq x \leq 1$.

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