Skip to content# Quadratic Inequalities: A Step-by-Step Guide to Solving by Factorization

## What are Quadratic Inequalities?

## Quadratic Inequalities Solving by Factorization

### Practice Questions

**Solution:**

## Exam Tips...

## Worksheet on Quadratic Inequalities

**Quadratic Inequalities**

- Quadratic inequalities are mathematical statements that combine quadratic equations and an inequality symbol.
They are essential for students studying algebra, as they frequently appear in exams and real-life situations.

In this article, we will discuss:

**What are Quadratic Inequalities?****Quadratic Inequalities Solving by Factorization?**

Here is one more link to practice a few extra questions: Maths Genie Quadratic Inequalities

- A quadratic inequality can have one of three forms:

**ax ^{2} + bx + c < 0, ax^{2} + bx + c > 0**

**or **

**ax ^{2} + bx + c ≤ 0, ax^{2} + bx + c ≥ 0**

where a, b, and c are real numbers.

- To solve quadratic inequalities by factorization, we need to follow these steps:

- Rewrite the inequality as an equation by replacing the inequality symbol with an equal sign.
- Factorize the quadratic equation.
- Determine the values of x that satisfy the equation.
- Use these values to solve the inequality.

- Solving quadratic inequalities by factorisation is a common technique used to find the values of x that satisfy the inequality.
- Here are the steps for solving a quadratic inequality using factorisation:

**STEPS for Solving Quadratic Inequalities:**

**Step #1:**Factorise the quadratic expression Start by factoring the quadratic expression on the left side of the inequality. For example, to solve the inequality

**x ^{2} – 4x + 3>0,** we can factor it as

**Step #2:**Find the critical points The critical points are the values of x that make the left side of the inequality equal to zero. In this example, the critical points are

**x = 1 and x = 3,**

which are the roots of the factored quadratic expression.

**Step #3:**Plot the critical points on a number line Draw a number line and mark the critical points on it. In this example, we mark

**x = 1 and x = 3 **

on the number line.

**Step #4:**To determine which intervals satisfy the inequality, you should test the intervals between the critical points.In each interval, pick any value and substitute it into the original inequality.

If the inequality is true for that value, then the interval satisfies the inequality; otherwise, it does not.

For example, if you want to test the interval x < 1, you can pick a value like x = 0 and substitute it into the original inequality.

Calculate:

**(0 – 1)(0 – 3), which results in (0 – 1)(0 – 3) > 0.**

Since this inequality is true, it indicates that the interval x < 1 satisfies the inequality.

**Step #5:**Write the solution set Finally, we can write the solution set for the inequality.- In this example, the solution set is

**x<1 or x>3.**

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**Solved Example: Quadratic Inequalities**

**Question 1:**Solve the inequality 2x^{2 }– 5x + 3 < 0 by factorisation.

Solution:

**Step #1:**Factorise the quadratic expression.

**2x ^{2 }– 5x + 3 = (2x – 3)(x – 1)**

**Step #2:**Find the critical points.- The critical points are

**x = 3/2 and x = 1, **

which are the roots of the factored quadratic expression.

**Step #3:**Plot the critical points on a number line. Mark x = 3/2 and x = 1 on the number line.

**Step #4:**Test the intervals.

Test the interval x<3/2 by picking x=1 and substituting it into the original inequality:

**(2(1) ^{2}-5(1)+3)<0, **

which is false.

Therefore, the interval x<3/2 does not satisfy the inequality.

Test the interval 3/2<x<1 by picking x=5/4 and substituting it into the original inequality:

**(2(5/4) ^{2 }**

which is true.

Therefore, the interval 3/2<x<1

Question 1: Solving the Quadratic Inequality:** x ^{2} - 5x + 6 > 0**

**Step #1:**Factor the quadratic expression:

**(x - 2)(x - 3) > 0**

**Step #2:**Set each factor equal to zero:

**x - 2 > 0 and x - 3 > 0**

**Step #3:**Solve each equation separately:

**For x - 2 > 0, x > 2**

**For x - 3 > 0, x > 3**

**Step #4:**Taking the intersection of the solutions, we get:

**x > 3**

So, the solution set is x > 3.

Question 2: Determining the Solution Set for **2x ^{2} - 3x ≤ 1**

**Solution:**

**Step #1:**Factor the quadratic expression:

**(x - 1)(2x + 1) ≤ 0**

**Step #2:**Set each factor equal to zero:

**x - 1 ≤ 0 and 2x + 1 ≤ 0**

**Step #3:**Solve each equation separately:

**For x - 1 ≤ 0, x ≤ 1**

**For 2x + 1 ≤ 0, 2x ≤ -1, x ≤ -1/2**

**Step #4:**Taking the union of the solutions, we get:

**x ≤ -1/2**

So, the solution set is x ≤ -1/2.

- It’s important to remember that when multiplying or dividing both sides of an inequality by a negative number, we need to reverse the inequality symbol.

**Question 1:** Finding the Values of x that Satisfy **-x ^{2} + 4x - 3 < 0**

**Question 2:** Solve the equation:** 2x ^{2} - 5x + 2 < 0.**

**Question 3:** Determining the Solution Set for ** x ^{2} + 9x + 20 > 0.**

**Question 4:** Finding the Values of x that Satisfy **-2x ^{2} + 5x - 2 ≤ 0.**

**Question 5:** Solve the equation:** 4x ^{2} - 16x + 15 < 0.**