Quadratic Inequalities
2026-02-28 11:08 Diff
https://brightchamps.com/en-us/math/algebra/quadratic-inequalities

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Last updated on October 28, 2025

A quadratic inequality compares a quadratic expression ax² + bx + c (where a ≠ 0) to zero using >, <, ≥, ≤ signs. They allow us to analyze intervals between roots and help determine if a given condition holds. Students should rewrite the inequality in standard form for solving, find the roots of ax² + bx + c = 0, and test sign intervals.

What are Quadratic Inequalities?

What Is Algebra? 🧮 | Simple Explanation with 🎯 Cool Examples for Kids | ✨BrightCHAMPS Math

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  • \(x^2 + x − 1 > 0 \)
  • \(2x^2 − 5x − 2 \)
  • \(x^2 + 2x − 1 < 0 \)

What are the Types of Quadratic Inequalities?

Quadratic inequalities compare a quadratic expression to zero, using inequality symbols: greater than, less than, greater than or equal to, and less than or equal to.

They are solved by finding roots and testing the signs of the expression over different intervals.

Type General Form Name Standard \(ax^2 + bx + c ≷ 0\) Positive/Negative/0m/0 No real roots \(<0\) Always positive or negative Repeated roots \(=0 \) Touches the x-axis once Factored (simple/repeated) \((x - a)(x - b) . . .≷0\) Linear and repeated linear factors Rational \(\frac{P(x)}{Q(x)} \gtrless 0 \) Rational quadratic inequalities Compound quad ≷ m quad Two-sided/bounded inequality

How To Solve Quadratic Inequalities

Step 1 -  Rewrite the equation to express the inequality:

Example: \(x^2 − 5x + 6 > 0\)

Step 2 - Find roots by factoring or formula:

\((x−2)(x−3)>0\) → roots: 2, 3.

Step 3 - Create intervals:

\((−∞,2),  (2,3),  (3,∞)\)

Step 4 - Test each interval by plugging in a sample value:

  • \(x = 0\): positive → true
     
  • \(x = 2.5 \): negative → false
     
  • \(x = 4 \): positive → true

Step 5. Write a solution:

All \(x∈ (−∞,2)  (3,∞) \)

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Notations Used In Quadratic Inequalities.

The symbols > and < (greater than, less than) replace “=” in a quadratic equation to form a quadratic inequality. The format is
\(ax² + bx + c > 0 \)  or   \(ax² + bx + c < 0\).

  • ( ) → Open brackets
  • [ ] → Closed brackets
  • o → Open value (x cannot take this value)
  • • → Closed value (x can take this value)

Examples:

  • (-1, 1) → x cannot take -1 and 1
  • [-1, 1) → x can take -1 but not 1
  • (-1, 1] → x cannot take -1 but can take 1
  • [-1, 1] → x can take both -1 and 1

What are the methods to solve Quadratic Inequalities?

Different methods used to solve quadratic inequalities are listed below:

Factoring + Test Intervals

  1. Rewrite as \(ax² + bx + c ≷ 0\).
     
  2. Factor & find roots.
     
  3. Divide the number line into intervals.
     
  4. Test each interval.

Example:

\(x^2 − 5x + 6 > 0  ⟹  (x − 2)(x − 3) > 0\)

\((−∞,2), (2,3), (3,∞) \)

Testing the values in the expression, we get, \(x = 0 \): true;\( x = 2.5 \): false; \(x = 4 \): true.

Solution: \(x∈(−∞,2)∪(3,∞) \)

Sign Chart Method

  1. Factor the expression.
     
  2. Mark the roots on a number line.
     
  3. Determine the sign of each factor in each region (alternating sign from rightmost interval).
     
  4. Combine to find where the product matches the inequality.

Example:

\(\frac{P(x)}{Q(x)} \gtrless 0 \)

\(x - 3 = 0, x = 3 \)

\(x + 2 = 0, x = -2 \)

So, the expression changes sign between \(x = -2\) and \(x = 3\)

Now, set intervals using roots to divide the number line:

\((-∞,-2), (-2,3), (3,∞)\)

Test each interval

\(x = -3: (x - 3) (x + 2) = (+)\)

\(x = 0: (x - 3) (x + 2) = (-) \)

\(x = 4: (x - 3) (x + 2) = (+)\)

The only region where the expression is negative is \((-2, 3)\)
So, \(x∈(-2,3)\)

Solution: \(x∈(-2,3)\)

Graphical Method

  1. Rewrite as \(y = ax2 + bx + c and y ≷ 0\)
     
  2. Sketch the parabola (upwards if \(a>0\), down if \(a<0\))
     
  3. Read off where the parabola is above/below the x-axis.

Example:

\(x^2 + 5x + 6 ≥ 0 \)

The graph crosses at \(x = −2\) and \(−3\) and opens upward since the linear coefficient is positive.

So, the expression is \(≥ 0\) outside \([−3,−2]\).

Solution: \(x∈(−∞, −3]∪[−2,∞)\).

Tips and Tricks to Master Quadratic Inequalities

The given tips and tricks help students understand and work with quadratic inequalities efficiently.
 

  • Factorize first – Convert the quadratic equation into its factored form to easily find the roots.
     

  • Mark critical points – Plot the roots on a number line to divide intervals for testing signs.
     

  • Test intervals – Check the sign of each interval to determine where the inequality holds true.
     

  • Remember parabola direction – Use the coefficient of \(x^2\) to know if the parabola opens upward or downward.
     

  • Use clear notation – Write your final answer in interval form or inequality form for clarity.

Real-Life Applications of the Quadratic Inequalities

The quadratic inequalities model helps us understand and solve real-world problems in the fields of physics, engineering, economics, and biology. Some of its applications are listed below:

  • Bridge design: This is useful to ensure the structural integrity of bridges. For example, civil engineers need to examine the maximum load a bridge can bear to determine the range of weight distribution that is safe. This helps in preventing breakdown.
  • Profit Margins in Business: Businesses use Quadratic Inequalities for modelling and analyzing the business profit margins. For example, a company uses it to decide the range of product prices that will create a desired profit level.
  • Radio Telescope Design: Radio telescopes use parabolic mirrors to focus incoming radio waves. This helps design the shape and dimensions of these mirrors to ensure optimal signal reception and focusing. 
  • Determining Safe Speed: This is useful in transportation, as it helps to find the Maximum safe speed on a curved and dangerous road map. Making sure that the vehicle can navigate the curve without skidding or losing control.
  • Area Calculation: This can be used to find the dimensions of a shape, like a rectangle, that meets a specific area requirement. For example, to find the dimensions of a garden or a plot that has the minimum required area.

Common Mistakes of the Quadratic Inequalities and How to Avoid Them

Quadratic inequalities are challenging for some students, and common mistakes can create incorrect answers. Recognizing and avoiding these errors is important for perfect problem-solving.

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Problem 1

For which values of x is x² − 4x + 3 > 0?

Okay, lets begin

x∈ (-∞,1) ∪ (3,∞).

Explanation

(x - 1)(x - 3) 3 > 0. Roots: x = 1, 3.

(-∞,1), (1,3), (3,∞).

x = 0: (0 - 1)(0 - 3) = (-1)(-3)  3 > 0, true

x = 2: (2 - 1)(2 - 3) = 1 · (-1) = -1 > 0, false

X = 4: (4 - 1)(4 - 3) = 3·1 = 3 > 0, true

Well explained 👍

Problem 2

Solve x² + 2x − 15 ≤ 0.

Okay, lets begin

−5 ≤ x ≤ 3.

Explanation

Factor to (x + 5)(x − 3).

The quadratic is non-positive between the roots.

Well explained 👍

Problem 3

Determine x such that 3x² − 12 ≤ 0.

Okay, lets begin

−2 ≤ x ≤ 2

Explanation

Rewrite as 3(x2 − 4) ≤ 0.

Hence, −2 ≤ x ≤ 2.

Well explained 👍

Problem 4

Find all x satisfying 2x²+ 7x + 6 < 0

Okay, lets begin

−2 < x < −3/2

Explanation

Factor (2x + 3)(x + 2) < 0.

The product is negative between the roots: −2 < x < −1.5.

Well explained 👍

Problem 5

Solve 4x² − 4x+ 1 ≥ 0.

Okay, lets begin

All real x.

Explanation

This is (2x − 1)2, which is always ≥ 0 for every x.

Well explained 👍

FAQs of the Quadratic Inequalities

1.What is a quadratic inequality?

A quadratic inequality is an inequality involving a quadratic expression, such as ax2 + bx + c, where the inequality symbol is <, ≤, >, or ≥.

2.How do I solve a quadratic inequality?

To solve, factor the quadratic expression, find the roots, determine intervals, test points within those intervals, and identify where the inequality holds.

3.What is the significance of the sign of a in ax² + bx + c?

The sign of a determines the direction of the parabola: if a > 0, it opens upwards; if a < 0, it opens downwards.

4.How do I handle strict inequalities < or >?

For strict inequalities, the solution does not include the boundary points; use open intervals to represent the solution set.

5.How do I express the solution set?

Solution sets can be expressed in interval notation, such as (−∞,−3)∪(2,∞), or in set-builder notation.

6.How can I help my child understand quadratic inequalities at home?

Use graphs to show how a parabola lies above or below the x-axis for different values of 
𝑥
x. Visualizing helps your child see which regions satisfy the inequality.

7.What are some real-life examples of quadratic inequalities I can use to explain the concept?

Examples include finding safe speed limits for turns, determining profit ranges in business, or calculating projectile heights that stay below a certain level.