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Quadratic Inequalities

By Tina Goosz

A quadratic inequality is an equation of the second degree that uses an inequality symbol in place of an equals symbol. The solutions to a quadratic inequality provide the two roots.

Why is this concept useful?

We use this concept in high school algebra, it can provide us information about what’s on either side of zero (positives and negatives).

Where does this concept fit into the curriculum?

High School

How can we use the concept:

Quadratic inequalities can be solved by the factorization method or by using the quadratic formula.

Quadratic formula: x =

−b±(b2−4ac)2a\frac{-b ± \sqrt(b^2-4ac)}{2a}

Steps to solve.

Step 1: Write the quadratic inequality in standard form:

ax2ax^{2}
+ bx + c where a, b and are coefficients and a ≠ 0.

Step 2: Determine the roots of the inequality.  In quadratic inequalities, these can also be called critical values.

Step 3: Write the solution in inequality notation or interval notation. If the quadratic inequality is in the form: (x – a) (x – b) ≥ 0, then a ≤ x ≤ b, and if it is in the form :(x – a) (x – b) ≤ 0, when a < b then a ≤ x or x ≥ b.

You can also use a graph to identify the solution set of the inequality.

Sample Math Problems

Question

x2−2x−3<0x^{2}-2x-3 < 0

KaTeX can only parse string typed expression
Answer

Step 1: Write the quadratic inequality in standard form:

ax2ax^2
+ bx + c where a, b and are coefficients and a ≠ 0.

This step is complete. 

Step 2: Determine the roots of the inequality.  In quadratic inequalities, these can also be called critical values.

(x - 3)(x + 1) < 0

Step 3: Write the solution in inequality notation or interval notation. If the quadratic inequality is in the form: (x – a) (x – b) ≥ 0, then a ≤ x ≤ b, and if it is in the form :(x – a) (x – b) ≤ 0, when a < b then a ≤ x or x ≥ b.

-1 < x < 3

Question

x2x^{2}
+ 5x + 6 ≥ 0

Answer

Step 1: Write the quadratic inequality in standard form:

ax2ax^{2}
+ bx + c where a, b and are coefficients and a ≠ 0.

This step is completed. 

Step 2: Determine the roots of the inequality.  In quadratic inequalities, these can also be called critical values.

x2x^{2}
+ 5x + 6 ≥ 0

(x + 3)(x + 2) ≥ 0

Step 3: Write the solution in inequality notation or interval notation. If the quadratic inequality is in the form: (x – a) (x – b) ≥ 0, then a ≤ x ≤ b, and if it is in the form :(x – a) (x – b) ≤ 0, when a < b then a ≤ x or x ≥ b.

-3 < x < -2

Question

x2x^{2}
< -7x - 10

Answer

Step 1: Write the quadratic inequality in standard form:

ax2ax^{2}
+ bx + c where a, b and are coefficients and a ≠ 0.

x2x^{2}
+ 7x + 10 < 0

Step 2: Determine the roots of the inequality.  In quadratic inequalities, these can also be called critical values.

(x + 2)(x + 5) < 0

Step 3: Write the solution in inequality notation or interval notation. If the quadratic inequality is in the form: (x – a) (x – b) ≥ 0, then a ≤ x ≤ b, and if it is in the form :(x – a) (x – b) ≤ 0, when a < b then a ≤ x or x ≥ b.

You can also use a graph to identify the solution set of the inequality.

x < -2 or x < -5

Question

x2x^{2}
- 11x + 30 ≤ 0

Answer

Step 1: Write the quadratic inequality in standard form:

ax2ax^{2}
+ bx + c where a, b and are coefficients and a ≠ 0.

This step is completed.

Step 2: Determine the roots of the inequality.  In quadratic inequalities, these can also be called critical values.

(x - 5)(x - 6) ≤ 0

Step 3: Write the solution in inequality notation or interval notation. If the quadratic inequality is in the form: (x – a) (x – b) ≥ 0, then a ≤ x ≤ b, and if it is in the form :(x – a) (x – b) ≤ 0, when a < b then a ≤ x or x ≥ b.

You can also use a graph to identify the solution set of the inequality.

x ≥ 5 or x ≥ 6

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Practice Math Problems

1.

x2x^{2}
- 8x - 16 > 0

2.

x2x^{2}
< -9x - 14

3. 25 + 10x ≥ -

x2x^{2}

4.

x2x^{2}
- x - 12 ≤ 0

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