Graphing Quadratic Inequalities

When it comes to understanding the behavior of curves, graphing quadratic equations and finding their roots is a helpful technique. Now, let's take what we've learned and apply it to graphing quadratic inequalities.

Defining Quadratic Inequalities

A quadratic inequality is a second-degree polynomial expression that uses an inequality sign instead of an equal sign. This lesson will cover the four types of quadratic inequalities (in two variables).

One Variable Quadratic Inequalities

A quadratic inequality in one variable has only one unknown in the quadratic expression and can be graphed on a single axis or number line. The graph represents all ordered pairs (x, y) that satisfy the inequality.

On the other hand, a quadratic inequality in two variables describes a region in the Cartesian plane with a parabola as the boundary curve, including both the x and y-axis.

Standard Forms of Quadratic Inequalities

The standard forms of quadratic inequalities (in one variable) are:

• ax2 + bx + c < 0
• ax2 + bx + c > 0
• ax2 + bx + c <= 0
• ax2 + bx + c >= 0

Solving Quadratic Inequalities In One Variable

The process for solving quadratic inequalities in one variable is similar to solving quadratic equations, except we are looking for a range of real numbers that satisfy the inequality instead of equating the expression to zero.

Basic factorization methods can be used to solve these inequalities. The steps are as follows:

• Step 1: Rewrite the quadratic inequality in general form, with ax2 + bx + c, where a ≠ 0, on one side of the inequality.
• Step 2: Completely factorize the quadratic expression.
• Step 3: Solve the corresponding equation to identify the roots of the inequality.
• Step 4: Determine the behavior of the inequality based on the roots.
• Step 5: Express the solution in interval or inequality notation.

Graphing Quadratic Inequalities In One Variable

Graphing quadratic inequalities in one variable can also be done using the graph of the given polynomial. Refer to the table below for the graphical representation of each type of inequality, with the shaded region representing the solution to the inequality.

Example 1

Let's solve the inequality x2 + 2x – 48 > 0.

Solution:

• Step 1: Rewrite the inequality as x2 + 2x – 48 > 0.
• Step 2: Factorize the quadratic expression to (x + 8)(x – 6) > 0.
• Step 3: Solve for the roots by setting (x + 8)(x – 6) = 0, which gives x = -8 or x = 6.
• Step 4: Since the inequality is greater than 0, we use Case 2 from the table, where a > 0. This means we need to choose the values of x for which the curve is above the x-axis.
• Step 5: Writing the solution in interval notation, we get (-8, 6).

Graph:

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