Quadratic Inequalities

In this lesson, 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.

1 < x < 3

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.

or x > 5

Videos

Solving quadratic inequalities -
Professor Edward Burger explains solving quadratic inequalities.

Solving quadratic inequalities - another example
Professor Edward Burger explains another example of solving quadratic inequalities.

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