7.1: Linear Inequalities
This activity is intended to supplement Algebra I, Chapter 6, Lesson 3.
ID: 8773
Time required: 60 minutes
Activity Overview
In this activity, students first look at tables of values to see that inequalities (in a single variable) are true for some values of the variable and not for others. They are guided to connect “true” with a value of 1 and “false” with a value of 0, creating a bridge from the table of values to a graph of the linear inequality. In Problem 2, students solve one-step linear inequalities using addition or subtraction and compare the graph of the original inequality with that of the simplified form, finding they are the same. In Problem 3, students perform a similar exploration solving one step linear inequalities using multiplication or division.
Topic: Linear Inequalities
- Graph a simple linear inequality of the form \begin{align*}x < a, \ x \le a, \ x > a\end{align*}, or \begin{align*}x \ge a\end{align*} on the real line.
- Use addition or subtraction to solve a “one step” linear inequality in a single variable.
- Use multiplication or division to solve a “one step” linear inequality in a single variable.
Teacher Preparation and Notes
- This activity is designed for use in an Algebra 1 or Pre–Algebra classroom.
- Prior to beginning the activity, students should download the LINEQUA program to their calculators. They should also have some experience solving simple one-step and multiple-step linear equations.
- This activity is intended to be a combination of teacher-led and student-centered activity. It is recommended that you introduce each problem by guiding students through the step-by-step instruction in a whole-class setting, and then allow them to complete the exercises individually or in small groups with your assistance.
- If time constraints prevent you from completing the activity in one class period, you may choose to complete Problem 1 one day and Problems 2 and 3 the next day.
- To download the calculator program, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=8773 and select "LINEQUA.8xp".
Associated Materials
- Student Worksheet: Linear Inequalities, http://www.ck12.org/flexr/chapter/9616
- LINEQUA.8xp (program)
Introduction
An inequality is a mathematical sentence that shows the relationship between two quantities using these signs: \begin{align*}>, \ge , <, \le,\end{align*} or \begin{align*}\neq\end{align*}.
Solving inequalities in a single variable is similar to solving equations in a single variable, but there are some important differences. One big difference occurs when you multiply or divide by a negative number. In this activity, you will practice solving linear inequalities and explore these differences using algebraic and graphing techniques.
Problem 1 – Table of Values
First students will use the LINEQUA program to explore the table of values of a simple inequality: \begin{align*}x \ge 4\end{align*}. Directions are given on the student worksheet for how to enter the inequality. Students will enter the left side, enter the right side, select a symbol, and choose to view the table. They can either press the number associated with each choice or use the arrow keys to move to the choice and press enter.
When students first view the table, they will only see \begin{align*}X, \ Y1\end{align*}, and \begin{align*}Y2\end{align*}. The first column \begin{align*}X\end{align*} shows the values of the variable, \begin{align*}x.\end{align*} The second column, labeled \begin{align*}Y1\end{align*}, shows the value of the left side for each \begin{align*}x-\end{align*}value. The third column, labeled \begin{align*}Y2\end{align*}, shows the value of the right side for each \begin{align*}x-\end{align*}value.
Students can use the right arrow key to move to the fourth column \begin{align*}Y3\end{align*}. Each entry in this column is either \begin{align*}a \ 1\end{align*} or \begin{align*}a \ 0\end{align*}. After investigating the second inequality \begin{align*}x < -2\end{align*}, students should realize that \begin{align*}a \ 0\end{align*} represents when the inequality is false, and a \begin{align*}1\end{align*} represents when the inequality is true.
Problem 2 – Graphing
Students are to graph the inequalities \begin{align*}x > 2\end{align*} and \begin{align*}x \ge 2\end{align*} one at a time, and then describe the differences they see. This is a preliminary investigation into open and closed circles.
Examples
The following examples show how to sketch the graph of an inequality on paper. Explain the two inequalities to students before doing the exercises.
\begin{align*}x < 5\end{align*}
The open circle at \begin{align*}5\end{align*} means that \begin{align*}x = 5\end{align*} is not a solution to the inequality. Since \begin{align*}x\end{align*} is less than \begin{align*}5\end{align*}, shade to the left of the circle.
\begin{align*}w \ge 4\end{align*}
The closed circle at \begin{align*}2\end{align*} means that \begin{align*}x = 2\end{align*} is a solution to the inequality. Since \begin{align*}w\end{align*} is greater than or equal to \begin{align*}5\end{align*}, shade to the right of the circle.
Problem 3 – Solving inequalities using addition and subtraction
Equivalent inequalities are inequalities with the same solutions. For example, \begin{align*}x - 3 > 5\end{align*} and \begin{align*}x > 8\end{align*} are equivalent inequalities. Explain to students that they can add or subtract the same number from both sides of an inequality without affecting the solutions, just as they can do with equations.
Examples
\begin{align*}x-3&>5 && \quad \ \ x+6 \le 10\\ x-3+3&>5+3 && x+6-6 \le 10-6\\ x&>8 && \qquad \quad \ x \le 4\end{align*}
Students will compare the starting inequality and the final inequality using the Compare Ineq choice in the LINEQUA program. The first inequality entered will appear in the top half of the screen. The second inequality entered will appear in the bottom half of the screen.
Caution: In some graphs, the open circle will appear to be filled in. This is because of the size of the pixels on the graph screen. For this reason, a “closed circle” is shown as a cross, and an “open circle” as a dot.
Problem 4 – Solving inequalities using mulitplication and division
Students are to graph four different pairs of inequalities that involve multiplication or division. This will lead them an investigation of when the inequality symbol is flipped or reversed.
Solutions
Problem 1
1. The numbers in the \begin{align*}Y1\end{align*} column are the same as the \begin{align*}x-\end{align*}values because the left side of the inequality is \begin{align*}x\end{align*}.
2. The numbers in the \begin{align*}Y2\end{align*} column are not affected by the \begin{align*}x-\end{align*}values, because the right side of the inequality is a constant, \begin{align*}4\end{align*}.
3. For all of the \begin{align*}x-\end{align*}values greater than or equal to \begin{align*}4\end{align*}
4. Sample: \begin{align*}x = 5 \rightarrow x \ge 4 \rightarrow 5 \ge 4\end{align*}; yes
5. For all of the \begin{align*}x-\end{align*}values less than \begin{align*}4\end{align*}.
6. Sample: \begin{align*}x = 3 \rightarrow x \ge 4 \rightarrow 3 \ge 4\end{align*}; no
7. For all \begin{align*}x-\end{align*}values less than \begin{align*}-2\end{align*}.
8. Sample: \begin{align*}x = -4 \rightarrow x < -2 \rightarrow -4 < -2\end{align*}; yes
9. For all \begin{align*}x-\end{align*}values greater than or equal to \begin{align*}-2\end{align*}.
10. Sample: \begin{align*}x = -1 \rightarrow x < -2 \rightarrow -1 < -2\end{align*}; no
11. a) \begin{align*}1\end{align*}
b) \begin{align*}0\end{align*}
Problem 2
12. The graphs of \begin{align*}x > 2\end{align*} has an open circle at \begin{align*}x = 2\end{align*}, and the graph of and \begin{align*}x \ge 2\end{align*} has a cross at \begin{align*}x = 2\end{align*}.
13. a) \begin{align*}t > 5\end{align*}
b) \begin{align*}p < -2\end{align*}
c) \begin{align*}z \ge -2\end{align*}
d) \begin{align*}y \le 0\end{align*}
Problem 3
14. a) \begin{align*}f \ge 7\end{align*}
b) \begin{align*}-2 > g\end{align*}
c) \begin{align*}u \le 4\end{align*}
d) \begin{align*}-7 < v\end{align*}
e) \begin{align*}-4 > h\end{align*}
f) \begin{align*}-6 \le t\end{align*}
Problem 4
15. a) yes; they have the same solutions
b) yes
16. a) yes; they have the same solutions
b) yes
17. a) no; they do not have the same solutions
b) no
18. a) yes; they have the same solutions
19. a) \begin{align*}\le\end{align*}
b) \begin{align*}>\end{align*}
c) \begin{align*}<\end{align*}
d) \begin{align*}\ge\end{align*}
20. a) \begin{align*}c \ge 4\end{align*}
b) \begin{align*}-8 > d\end{align*}
c) \begin{align*}w \le -3\end{align*}
d) \begin{align*}-4 < x\end{align*}
e) \begin{align*}d<-\frac{2}{3}\end{align*}
f) \begin{align*}g < 7\end{align*}
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