1. Lesson 8

1.6. Explore 2

Mathematics 20-1 Module 4

Module 4: Quadratic Equations and Inequalities

 

Solving Quadratic Inequalities in One Variable

 

A quadratic inequality in one variable can be written in any one of the following forms where a, b, and c are real numbers and a ≠ 0:

  • ax2 + bx + c < 0
  • ax2 + bx + c ≤ 0
  • ax2 + bx + c > 0
  • ax2 + bx + c ≥ 0

In Lesson 7, you explored inequalities in two variables. Quadratic inequalities in one variable may be solved by referring to the graphs of corresponding quadratic inequalities in two variables. The following inequalities in two variables correspond to the preceding inequalities in one variable:

  • ax2 + bx + c < y
  • ax2 + bx + c ≤ y
  • ax2 + bx + c > y
  • ax2 + bx + c ≥ y

In Discover you may have noticed something about the solution set of an inequality in one variable. The solution set corresponds to certain x-values of the graph of the corresponding inequality in two variables. Depending on the inequality, the chosen x-values will lie above, on, or below the x-axis. Quadratic inequalities in one variable can be solved by both graphical and algebraic methods.

The following example shows how you can solve quadratic inequalities in one variable by analyzing the graphs of corresponding functions in two variables. This example is based on the same question posed in Try This 1 question 4. Retrieve your work from Try This 1 so that you can compare your solution to the solution in the example.

 

Example

 

Solve x2 − 2x − 8 ≥ 0.

 

Solution

 

Graph the corresponding function y = x2 − 2x − 8.

 

 

This shows the graph of the parabola y = x squared – 2x – 8. It crosses the x-axis at (-2, 0) and (4, 0) and the y-axis at (0, -8).

 

The problem asks for the solutions to x2 − 2x − 8 ≥ 0. Since the function that the graph represents is y = x2 − 2x − 8, you need to determine where y ≥ 0. You can do this by identifying the points on the graph where the y-coordinate is greater than or equal to 0. These points are on and above the x-axis, shown in red on the following graph.

 

 

This shows the graph of the parabola y = x squared – 2x – 8. It crosses the x-axis at (-2, 0) and (4, 0) and the y-axis at (0, -8). The parts of the curve above the x-axis are red.

 

You can report the solution as the domain of the parts of the graph highlighted in red. The domain is all real values less than or equal to –2 or greater than or equal to 4.

 

The solution set is {x | x ≤ − 2 or x ≥ 4, x ∈ R}.

 

Compare your solution from Try This 1 to the solution in the example. Make any necessary revisions to your work.


textbook

Turn to “Example 1” on pages 478 to 479 of the textbook. You are asked to solve a different quadratic inequality. Work your way through Method 1 only. Ask yourself the following questions as you work through the method:

  • What other ways could you report the solution?

  • How can you sketch the graph of a quadratic function when the function is written in standard form?
Self-Check 1

 

Use the following information to answer the next two questions.

 

 

The graph of f(x) = –5x squared + 20x – 15 is shown. The graph has vertex at (2, 5) and crosses the x-axis at (1, 0) and (3, 0).

  1. What is the solution to the quadratic inequality −5x2 + 20x − 15 < 0?

    1. x < 1
    2. x > 3
    3. 1 < x < 3
    4. x < 1 or x > 3

    Answer

  2. What is the solution to the quadratic inequality −5x2 + 20x − 15 ≥ 0?

    1. 1 < x < 3
    2. 1 ≤ x ≤ 3
    3. x < 1 or x > 3
    4. x ≤ 1 or x ≥ 3

    Answer

  3. Select the choice or choices for which the stated value of x is a solution to the given inequality.

    1. x = 0 for −x2 − 5x + 1 < 0
    2. x = −3 for x2 + x − 6 > 0
    3. x = 12 for −0.5x2 + 4x + 17 ≥ 0
    4. x = 1 for 2x2 − 7x + 3 ≤ 0

    Answer

  4. The solution to the inequality 2x2 − 7x ≤ 3 is found in the interval a ≤ x ≤ b. What is the sum of a and b to the nearest tenth? Answer
Plot the x-intercept(s). Plot the y-intercept. Then determine the vertex. First, find the x-coordinate that is halfway between the x-intercepts. Next, substitute that x-coordinate into the function to determine the corresponding y-coordinate. Plot the vertex, and sketch the graph.
Substitute x-values into the function to see if the resulting answer meets the ≤ 0 requirement.