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Precalculus for Team-Based Inquiry Learning: 2024 Development Edition

TBIL Fellows

Section 1.7 Quadratic and Rational Inequalities (EQ7)

Subsection 1.7.1 Activities

Remark 1.7.1.

In Section 1.5 and Section 1.6 we learned how to solve quadratic and rational equations. In this section, we use these skills to solve quadratic and rational inequalities.

Activity 1.7.2.

Consider the quadratic inequality
\begin{equation*} x^2-4x-32\gt0\text{.} \end{equation*}
(a)
Use a graphing utility to graph the function \(f(x)=x^2-4x-32\text{.}\) Which part of the graph represents where \(x^2-4x-32\gt0\text{?}\)
(b)
Which pieces of information about \(f(x)=x^2-4x-32\) were needed to answer part (a)?
  1. The \(y\)-intercept
  2. The \(x\)-intercepts
  3. The minimum value
(c)
Use algebra to find the \(x\)-intercepts of \(f(x)=x^2-4x-32\) and mark them on a number line. Then, shade the part of the number line where \(x^2-4x-32>0\text{.}\)
(d)
Now use interval notation to express where \(x^2-4x-32>0\text{.}\)
  1. \(\displaystyle (-4,8)\)
  2. \(\displaystyle [ -4,8]\)
  3. \(\displaystyle (-\infty,-4)\cup(8,\infty)\)
  4. \(\displaystyle (-\infty,-4]\cup[8,\infty)\)

Definition 1.7.3.

A sign chart is a number line representing the \(x\)-axis that shows where a function is positive or negative. Instead of shading, which can be ambiguous, it is often decorated with a ’+’ or a ’-’ to indicate which regions are positive or negative. For example, a sign chart for \(f(x)=x^2-4x-32\) is below.
Figure 1.7.4. A sign chart for the function \(f(x)=x^2-4x-32\text{.}\)

Activity 1.7.5.

Solve the quadratic inequality algebraically
\begin{equation*} 2x^2-28 \lt 10x\text{.} \end{equation*}
Write your solution using interval notation.
  1. \(\displaystyle \left(-\infty,-2\right)\cup\left(7,\infty\right)\)
  2. \(\displaystyle \left(-\infty,-7\right)\cup\left(2,\infty\right)\)
  3. \(\displaystyle \left(-7,2\right)\)
  4. \(\displaystyle \left(-2,7\right)\)

Activity 1.7.6.

Solve the inequality
\begin{equation*} -2x^2-10x-10 \ge 6x+20\text{.} \end{equation*}
Write your solution using interval notation.
  1. \(\displaystyle [-5,-3]\)
  2. \(\displaystyle (-\infty,-5]\cup[-3,\infty)\)
  3. \(\displaystyle [3,5]\)
  4. \(\displaystyle (-\infty,3]\cup[5,\infty)\)

Activity 1.7.7.

Consider the rational inequality
\begin{equation*} \frac{4x+3}{x+2} \gt x\text{.} \end{equation*}
(a)
For which of the following functions will a graph help us solve the rational inequality above?
  1. \(\displaystyle f(x)=\frac{4x+3}{x+2}\)
  2. \(\displaystyle g(x)=\frac{4x+3}{x+2}-x\)
  3. \(\displaystyle h(x)=x-\frac{4x+3}{x+2}\)
(b)
Use a graphing utility to graph the function \(g(x)=\frac{4x+3}{x+2}-x\text{.}\) Which part of the graph represents where \(\frac{4x+3}{x+2}-x\gt0\text{?}\)
(c)
Simplify \(\frac{4x+3}{x+2} - x\) into a single rational expression.
  1. \(\displaystyle \frac{4x+3}{x+2}\)
  2. \(\displaystyle \frac{3x+3}{x+2}\)
  3. \(\displaystyle \frac{x^2+6x+3}{x+2}\)
  4. \(\displaystyle \frac{-x^2+2x+3}{x+2}\)
(d)
How do these values you found in part (b) relate to the numerator and the denominator of the combined rational function in part (c)?
Hint.
Factor the numerator.
(e)
For what values is the original inequality a true statement?
  1. \(x \lt -2\) and \(-1 \lt x \lt 3\)
  2. \(-2 \lt x \lt -1\) and \(x \gt 3\)
  3. \(\displaystyle -2\lt x \lt -1\)
  4. \(\displaystyle 1 \lt x \lt 3\)
(f)
How can we express the answers to part (e) for the rational inequality using interval notation?
  1. \(\displaystyle \left(-\infty,-2\right) \cup (-1,3)\)
  2. \(\displaystyle \left(-2,-1\right) \cup (3,\infty)\)
  3. \(\displaystyle \left(-2,-1\right) \)
  4. \(\displaystyle \left(1,3\right) \)

Definition 1.7.8.

The values on the \(x\)-axis where a function is equal to zero or undefined are called partition values.

Activity 1.7.9.

Solve the rational inequality
\begin{equation*} \frac{x+8}{x-2} \le \frac{x+10}{x+5}\text{.} \end{equation*}
(a)
Write the solution using interval notation.
  1. \(\displaystyle (-\infty, -12)\cup[-5,2]\)
  2. \(\displaystyle (-\infty, -12]\cup(-5,2)\)
  3. \(\displaystyle (-12,-5]\cup[2,\infty)\)
  4. \(\displaystyle [-12,-5)\cup(2,\infty)\)
(b)
Compare the interval notation from Activity 1.7.7 to the interval notation for this activity. When do we include the partition values in the answer with a bracket?

Exercises 1.7.2 Exercises

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