# 7.3: Can I Graph You, Too?

**At Grade**Created by: CK-12

*This activity is intended to supplement Algebra I, Chapter 6, Lesson 6.*

## Probmem 1 - Introduction to Disjunction and Conjunction

Consider the equation \begin{align*}|x|=5\end{align*}. To solve, you would graph both sides of the equation as functions \begin{align*}(y =|x|\end{align*} and \begin{align*}y = 5\end{align*}) and mark the solution as the area where the graphs intersect.

The same method can be applied to inequalities.

Press **APPS** and select the **Inequalz** app. Press any key to begin.

**Example 1**: \begin{align*}|x|<5\end{align*}

- Using \begin{align*}Y1=\end{align*} graph the left side as \begin{align*}y = |x|\end{align*}. The absolute value function is located by pressing
**MATH**\begin{align*}\rightarrow\end{align*} and selecting**abs(**. - Using \begin{align*}Y2=\end{align*} graph the right side as \begin{align*}y < 5\end{align*}. On the equals sign, press
**ALPHA**\begin{align*}[F2]\end{align*} for the \begin{align*}<\end{align*} sign. Press**ZOOM**and select**ZoomStandard**. - Find the intersection points by pressing \begin{align*}2^{nd}\end{align*}
**[TRACE]**and selecting**intersect**. Now just move the cursor to the intersection point and press**ENTER**three times. The solution is where the shading overlaps the graph of the absolute value function.

In this case, the solution is \begin{align*}-5 < x < 5\end{align*}.

When an absolute value is less than a number, it is a **conjunction** because the solution is just one part of the graph.

\begin{align*}|ax+b| < c \quad \rightarrow \quad -c < ax + b < c.\end{align*}

**Example 2:** \begin{align*}|x-4| \ge 8\end{align*}

- Using \begin{align*}Y1=\end{align*} graph the left side as \begin{align*}y = |x-4|\end{align*}.
- Using \begin{align*}Y2=\end{align*} graph the right side as \begin{align*}y \ge 8\end{align*}. On the equals sign, press
**ALPHA**\begin{align*}[F5]\end{align*} for the \begin{align*}\ge\end{align*} sign. Press**WINDOW**to choose appropriate window settings. - Find the intersection points.

In this case, the solution is \begin{align*}x \le -4\end{align*} **or** \begin{align*}x \ge 12\end{align*}.

When an absolute value is greater than a number, it is a **disjunction** because the solution is two separate parts of the graph.

\begin{align*}|ax + b| > c \quad \rightarrow \quad ax + b < -c \quad \text{or} \quad ax + b > c.\end{align*}

## Problem 2 - Application of Disjunction and Conjunction

For the problems below, write the inequalities as either a conjunction or disjunction, then solve for \begin{align*}x\end{align*}. Check your solution by graphing using the method described in Examples 1 and 2. Please use your graphing calculator to check your results.

**#1:** \begin{align*}|2x-3|>9\end{align*}

**#2:** \begin{align*}\left |\frac{1}{3} x -10 \right | \le 11\end{align*}

**#3:** \begin{align*}|3x|-1 \ge 5\end{align*}

**#4:** \begin{align*}2 |4x-7| + 6 < 18\end{align*}

## Problem 3 - Real World Application

One application of absolute value inequalities is engineering tolerance. Tolerance is the idea that an ideal measurement and an actual measurement can only differ within a certain range.

A bolt with a \begin{align*}10 \ mm\end{align*} diameter has a tolerance range of \begin{align*}9.965 \ mm\end{align*} to \begin{align*}10 \ mm\end{align*}, while the hole that it fits into has a tolerance range of \begin{align*}10.05 \ mm\end{align*} to \begin{align*}10.075 \ mm\end{align*}.

How can you express the tolerances of both the bolt and the hole in terms of an absolute value inequality?