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Quadratic Inequalities


These free video lessons with examples and solutions help Algebra students learn how to solve quadratic-inequalities.




 
In these lessons, we will look at solving quadratic inequalities.

Example:

Solve the quadratic inequality x2 – 4x > –3

Solution:

Step 1: Make one side of the inequality zero

x2 – 4x > –3
x2 – 4x + 3 > 0

Step 2: Factor the quadratic expression

x2 – 4x + 3 > 0
(x – 3)(x – 1) > 0

Step 3: Find the range of values of x which satisfies the inequality.
(x – 3)(x – 1) > 0 (y is positive): we choose the interval for which the curve is above the x-axis.
x < 1 or x > 3

Note: If the quadratic inequality was (x – 3)(x – 1) < 0 (y is negative), we would have chosen the interval for which the curve is below the x-axis i.e. 1 < x < 3

The following graphs show the solutions for x2 – 4x + 3 > 0 and x2 – 4x + 3 < 0. Scroll down the page for more examples and solutions.

Solve Quadratic Inequalities




Example:

Solve 2x2 < 9x + 5

Solution:

Step 1: Make one side of the inequality zero

2x2 < 9x + 5
2x2 – 9x – 5 < 0

Step 2: Factor the quadratic expression

2x2 – 9x – 5 < 0
(2x + 1)(x – 5) < 0

Step 3: Find the range of values of x which satisfies the inequality.

(2x + 1)(x – 5) < 0 (y is negative): we choose the interval for which the curve is below the x-axis.


Note: If the quadratic inequality was (2x + 1)(x – 5) > 0 (y is positive) we would have chosen the interval for which the curve is above the x-axis i.e. or x > 5

How to solve quadratic inequalities?
Guidelines for solving Quadratic Inequalities
1. Find all the zeros of the polynomial, and arrange the zeros in increasing order. The zeros are called its critical numbers.
2. Plot those numbers on the number line as open or closed points based upon the original inequality symbol.
3. Choose a test value in each interval to see if the interval satisfies the inequality or not. If the test value produces a true statement, the entire interval will be true. If the interval produces a false statement, the entire interval is false.
4. Clearly graph your solution and state the solution using interval notation or inequalities.

Examples:
1. Solve x2 - 6x - 16 ≤ 0
2. Solve 2x2 - 11x + 12 > 0
3. Solve x2 + 4 > 0

Solve a Quadratic Inequality
Example:
x2 - x - 12 ≤ 0
How to solve a Quadratic Inequality?
Example:
2x2 + 3x - 5 > 0
Solving Quadratic Inequalities
1) Turn inequality into an equation.
2) Find solutions.
3) Make a number line, and check each solution and interval.
Examples:
Solve x2 + 2x - 8 ≥ 0


Solving Quadratic Inequalities - Step by step
Find all the solutions to
2x2 + 5x - 12 ≥ 0
Solving Quadratic Inequalities
Find all the solutions to
2x2 < -4x + 6

Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.


You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.


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