Quadratic Inequalities

Section 4.4 Quadratic Inequalities

In this lesson we solve quadratic inequalities graphically and algebraically, and write their solutions with interval notation. Note that our algebraic technique uses properties of the parabola; we do not use a "sign graph" to solve inequalities.

Activity 4.4.1. Equal, Less Than, Greater Than.

Here are three copies of the graph of \(~y=x^2-4\text{.}\)

On the first graph, we’ll solve the equation \(~x^2-4=0\text{.}\) (Remember that \(~y=x^2-4\text{.}\))
1. Find two points on the graph with \(y=0\text{.}\) Put dots there.
2. What are the \(x\)-coordinates of those points?
3. The solutions are:
On the second graph, we’ll solve the inequality \(~x^2-4 \lt 0\text{.}\)

Explain that because the graph shows points for which \(~y=x^2-4\text{,}\) the solutions to \(~x^2-4 \lt 0\) are just the \(x\)-coordinates of points on the graph with \(y \lt 4\text{.}\) You will need to go over this reasoning on every problem.
1. Mark all points on the graph that have \(y \lt 0\text{.}\)
2. On the \(x\)-axis, mark the \(x\)-coordinates of all those points.
3. Write an inequality to describe the portion of the \(x\)-axis marked.
On the third graph, we’ll solve the inequality \(~x^2-4 \gt 0\text{.}\)
1. Mark all points on the graph that have \(y \gt 0\text{.}\)
2. On the \(x\)-axis, mark the \(x\)-coordinates of all those points.
3. Write two inequalities to describe the portion of the \(x\)-axis marked.
  
  Emphasize that because the solution set consists of two intervals, we need two inequalities to describe it. Also explain why \(-2 \gt x \gt 2\) is not correct.

Activity 4.4.2. Solving Quadratic Inequalities Graphically.

Here are two copies of the graph of

\begin{equation*} ~y=175-18x-x^2~ \end{equation*}

Use the first graph to solve \(~175-18x-x^2 = 0~\text{.}\)

(Hint: Notice that one of the \(x\)-intercepts is \(25\text{.}\) Think of the factored form of the equation. What is the other factor?)

\begin{equation*} (x+25)(x- \fillinmath{XXX})=0 \end{equation*}

Start using the language "between the \(x\)-intercepts or outside the \(x\)-intercepts" or something similar to describe the choice of which part of the \(x\)-axis provides the solutions.
Use the second graph to solve \(~175-18x-x^2 \lt 0~\text{.}\)
1. Mark all points on the graph that have \(y \lt 0\text{.}\)
2. On the \(x\)-axis, mark the \(x\)-coordinates of all those points.
3. Write two inequalities to describe the portion of the \(x\)-axis marked.
Solve the inequality \(~20+4x-x^2 \le 8\text{.}\)
1. Rewrite the inequality so that the right side is zero.
  
  This problem also provides an opportunity to review graphing a parabola by hand. Notice that the coefficient of \(x^2\) is negative, so expect some confusion about finding the \(x\)-intercepts versus finding the \(x\)-coordinate of the vertex.
2. Graph the equation \(~y=12+4x-x^2\text{.}\)
  
  \(y\)-intercept:
  
  \(x\)-intercepts:
  
  (Solve \(~~12+4x-x^2=0)\)
  
  \begin{align*} \text{vertex:}~~~~~~x_v \amp = \dfrac{-b}{2a} = ~~~~~~~~~~~~~~~~\\ y_v \amp = ~~~~~~~~~~~~~~~~~ \end{align*}
3. Use the graph to solve the inequality. Write the solution with interval notation.

Activity 4.4.3. Solving Inequalities Algebraically.

To solve a quadratic inequality,

\begin{equation*} ax^2+bx+c \lt 0~~~~\text{or}~~~~ax^2+bx+c \gt 0 \end{equation*}

we must answer two questions:

What are the \(x\)-intercepts of the related graph?
Does the parabola open up or down?

Remember that you may need to use extraction of roots or the quadratic formula to find the \(x\)-intercepts of a parabola.

Use the graph to solve

\begin{equation*} ~x^2-4x-5 \gt 0~ \end{equation*}

Show the solution on the graph. Write your answer in interval notation.

Take a few minutes to give examples of interval notation. In this Lesson, we use \(\cup\) to indicate the union of two intervals, but we do not discuss intersection of sets.
Follow the steps to solve \(~4x^2+8x-5 \le 0\text{.}\)
Find the \(x\)-intercepts of \(~y = 4x^2+8x-5\text{.}\)

Does the graph of \(~y = 4x^2+8x-5~\) open up or down? Make a rough sketch of the graph, and label the \(x\)-intercepts.

Remind students that once we have found the \(x\)-intercepts as boundary points for our solution set, we only need to decide whether the solutions lie between the intercepts or outside them -- we don’t need any other information about the parabola (such as the vertex). So we only need to know if the parabola opens up or down.

Use the graph to solve the inequality. Write your answer in interval notation.
Solve the inequality \(~x^2 \ge 18\text{.}\)
Write the inequality in standard form.

Find the \(x\)-intercepts of the corresponding graph.

Make a rough sketch of the graph. Use the graph to solve the inequality. Write your answer with interval notation.
Problems (c) and (d) point out that sometimes we have to use extraction of roots or the quadratic formula to find the \(x\)-intercepts.
Solve the inequality \(~10-8x+x^2 \gt 4\text{.}\)
Write the inequality in standard form.

Find the \(x\)-intercepts of the corresponding graph, (Use the quadratic formula.)

Make a rough sketch of the graph. Use the graph to solve the inequality. Write your answer with interval notation.

Subsection 4.4.1 Check Your Understanding

In Activity 1, we saw that for any value of \(x\text{,}\) the value of \(x^2-4\) is either , , or .
Suppose you have a point \(P\) with \(y\)-coordinate 8 on the graph of an equation. How do you find the \(x\)-value that produces \(y=8\) in the equation?
When you solve a quadratic inequality with a graph, how do you find the boundary points of the solution interval(s)?
Use a graph to explain why the inequality \(~x^2+1 \lt 0~\) has no solution.

Subsection 4.4.2 Wrap-Up

In this Lesson, we worked on the following skills and goals related to quadratic models:

Write compound inequalities
Use interval notation
Solve quadratic inequalities graphically
Solve quadratic inequalities algebraically

Subsection 4.4.3 Questions for Writing or Discussion

Explain how to use a graph to find the \(x\)-value that produces a particular \(y\)-value for a quadratic equation.
Explain why you cannot write the solutions to \(x^2 - 4 \gt 0\) as a single inequality.
Explain how to use a graph to solve \(x^2-4x+77 \le 0\text{.}\)
Describe how to use interval notation.

Concept Questions.

What are the solutions of the inequality \(~x^2+1 \lt 0\text{?}\)
1. \(\displaystyle (-1,1)\)
2. \(\displaystyle (-\infty, -1) \cup (1, \infty)\)
3. All real numbers
4. No solutions
Which points on the parabola \(~y=ax^2+bx+c~\) do we need to know to solve the inequality \(~ax^2+bx+c \gt 0\text{?}\)
1. The \(x\)-intercepts
2. The \(y\)-intercept
3. The vertex
4. All of these
Why do we need to know whether tha parabola in question 2 opens up or down?
1. To decide whether to use \(\gt\) or \(\lt\) in the solution.
2. To decide whether the solutions lie between the \(x\)-intercepts or outside them.
3. To decide whether the solutions are positive or negative.
4. To help us find the \(x\)-intercepts.
What does the notation \([-3,3]\) mean?
1. The point with \(x\)-coordinate \(-3\) and \(y\)-coordinate \(3\text{.}\)
2. \(x=-3\) or \(x=3\)
3. All real numbers between \(-3\) and \(3\text{,}\) excluding the endpoints.
4. All real numbers between \(-3\) and \(3\text{,}\) including the endpoints.

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