7.3: Can I Graph You, Too?
This activity is intended to supplement Algebra I, Chapter 6, Lesson 6.
ID: 11479
Time Required: 30 minutes
Activity Overview
In this activity, students will explore absolute value inequalities graphically, numerically, and algebraically. They will rewrite absolute value inequalities as compound inequalities without absolute value and solve.
Topic: Absolute Value Inequalities
- Compound Inequalities
- Disjunctions and Conjunctions
Teacher Preparation and Notes
- Students should know how to graph linear inequalities by shading the appropriate half-plane. They should also know how to graph a linear absolute value function which produces a “v” shape.
- Teachers may want to give more guidance regarding isolating the absolute value expression on the left-hand side of the inequality before writing the disjunction or the conjunction.
- Students will be using the Inequality Graphing App in this activity.
- To download the calculator program, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=11479 and select "Inequality Graphing" under the Applications header.
Associated Materials
- Student Worksheet: Can I Graph You, Too?, http://www.ck12.org/flexr/chapter/9616, scroll down to the third activity.
- Inequality Graphing Application
Introduction to Disjunction and Conjunction
In this activity, students will explore absolute value inequalities graphically and numerically. Example 1 describes how students would graphically solve the equation \begin{align*}|x|<5\end{align*}. This may be a different approach than your students have seen, but it helps them visualize what is going on.
Teaching Notes:
You may want to further explain the terms conjunction and disjunction.
Students should notice that the solution is an interval, not \begin{align*}(x, \ y)\end{align*} points.
When using the Inequalz app, students can select the symbol that they need by pressing \begin{align*}a\end{align*} and then the key below where the symbol is shown. This will automatically shade the graph and display the horizontal line as dashed or solid.
Challenge the students to graph the inequality on a piece of paper and to draw a segment on the \begin{align*}x-\end{align*}axis representing the solution, like they would on a number line.
Make sure that they understand that for a disjunction, the solution goes to negative and positive infinity.
Application of Disjunction and Conjunction
Students will use what they’ve learned to solve inequalities graphically and algebraically. The absolute values are by themselves for the first two problems, but for the second two problems, students will need to get the absolute value by itself on the left hand side before writing as a conjunction or disjunction.
When confirming their answer by graphing, students should graph the left side and right side as they appear originally, not after getting the absolute value by itself.
The graphs and the resulting answers appear to the right and on the next page.
Problem 1: \begin{align*}x < -3\end{align*} or \begin{align*}x > 6\end{align*}
Problem 2: \begin{align*}-3 \le x \le 63\end{align*}
Problem 3: \begin{align*}x \le -2\end{align*} or \begin{align*}x \ge 2\end{align*}
Problem 4: \begin{align*}0.25 < x < 3.25\end{align*}
Real World Application
Students are introduced to the idea of engineering tolerance and how absolute value inequalities are used to express this concept. They are asked to express the given bolt and hole tolerances as absolute value inequalities.
To write these inequalities, students will have to find the median of the two values and distance from the median to an end value.
Student Answers
bolt: \begin{align*}|x-9.9825| \le 0.0175\end{align*}
hole: \begin{align*}|x-10.0625| \le 0.0125\end{align*}