7.1: Linear Inequalities
This activity is intended to supplement Algebra I, Chapter 6, Lesson 3.
Problem 1 – Table of Values
In this problem, you will explore and graph a simple inequality:
Enter the left side of the inequality,
Then enter the right side of the inequality,
Select an inequality symbol. Press
Choose 1:View Table to see a table of values.
The calculator displays a table with several columns. The first column
1. Describe the numbers in the
2. Describe the numbers in the
Now look at the fourth column
(Note: To scroll up or down, return to the
3. For what
4. Substitute one of these
5. For what
6. Substitute one of these
Press ENTER to exit the table, then press ENTER again to select 1:Another Ineq.
This time enter the inequality
Choose 1:View Table from the menu. Look at each column. Again, each entry in the
7. For what
8. Substitute one of these values in for
9. For what
10. Substitute one of these values in for
11. Complete each statement.
a) If the
b) If the \begin{align*}x-\end{align*}value makes the inequality false, the entry in the \begin{align*}Y3\end{align*} column is _______.
Problem 2 – Graphing
Now you are going to look at the graph of a simple inequality. Press ENTER to exit the table, and then press ENTER again to select 1:Another Ineq. Enter the inequality \begin{align*}x > 2\end{align*}. Choose 2:View Graph from the menu.
The calculator draws a line above the \begin{align*}x-\end{align*}values on the number line where the inequality is true. The inequality is not true when \begin{align*}x = 2\end{align*}, so an open circle is displayed there.
Press ENTER to exit the graph screen, and then press ENTER again to select 1:Another Ineq. Graph \begin{align*}x \ge 2\end{align*}.
12. Describe the difference between the graphs of \begin{align*}x > 2\end{align*} and \begin{align*}x \ge 2\end{align*}.
Exercises
13. Graph each inequality using your graphing calculator. Sketch the graphs here.
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 – Solving Inequalities using Addition and Subtraction
You can use your calculator to check that two inequalities are equivalent. To see that \begin{align*}x - 3 > 5\end{align*} and \begin{align*}x > 8\end{align*} are equivalent, run the LINEQUA program. Enter the inequality \begin{align*}x - 3 > 5\end{align*}.
Choose Compare Ineq. to compare this inequality to another.
Enter \begin{align*}x > 8\end{align*}. The calculator displays the graphs of \begin{align*}x - 3 > 5\end{align*} and \begin{align*}x > 8\end{align*} on the same screen. The graphs are the same, so the inequalities are equivalent.
- 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.
Exercises
14. Solve each inequality. Use your calculator to compare the original inequality with the solution. Then sketch the graph of the solution.
a) \begin{align*}f - 5 \ge 2\end{align*}
b) \begin{align*}-4 > g - 2\end{align*}
c) \begin{align*}u + 1 \le 5\end{align*}
d) \begin{align*}1 < 8 + v\end{align*}
e) \begin{align*}-5 > h - 1\end{align*}
f) \begin{align*}-5 \le 1 + t\end{align*}
Problem 4 – Solving Inequalities using Multiplication and Division
Use your calculator to compare the graphs of the given inequalities.
15. \begin{align*}\frac{x}{5} \le - 1\end{align*} and \begin{align*}x \le -5\end{align*}
a) Are these equivalent inequalities? Explain.
b) Can you multiply both sides of an inequality by \begin{align*}5\end{align*} without changing its solutions?
16. \begin{align*}4x > 8\end{align*} and \begin{align*}x > 2\end{align*}
a) Are these equivalent inequalities? Explain.
b) Can you divide both sides of an inequality by \begin{align*}4\end{align*} without changing its solutions?
17. \begin{align*}-x > 4\end{align*} and \begin{align*}x > -4\end{align*}
a) Are these equivalent inequalities? Explain.
b) Can you multiply both sides of an inequality by \begin{align*}-1\end{align*} without changing its solutions?
18. \begin{align*}-x > 4\end{align*} and \begin{align*}x < -4\end{align*}
a) Are these equivalent inequalities? Explain.
Exercises
19. Compare graphs to find the inequality symbol that makes each pair of inequalities equivalent.
a) \begin{align*}\frac{v}{-4} \ge 2 \quad v \underline{\;\;\;\;\;\;\;\;}-8\end{align*}
b) \begin{align*}- \frac{d}{3} < -3 \quad d \underline{\;\;\;\;\;\;\;\;} 9\end{align*}
c) \begin{align*}-2h > -2 \quad h \underline{\;\;\;\;\;\;\;\;} 1\end{align*}
d) \begin{align*}-5r \le 10 \quad r \underline{\;\;\;\;\;\;\;\;} -2\end{align*}
To solve an inequality using multiplication or division, multiply or divide both sides of the inequality by the same number. However, if you multiply or divide both sides by a negative number, you must reverse the inequality symbol to obtain an equivalent inequality.
20. Solve each inequality. Use your graphing calculator to compare the original inequality with the solution. Then sketch its graph.
a) \begin{align*}\frac{c}{4} \ge 1\end{align*}
b) \begin{align*}2 < - \frac{d}{4}\end{align*}
c) \begin{align*}3w \le -9\end{align*}
d) \begin{align*}20 > -5x\end{align*}
e) \begin{align*}18d < -12\end{align*}
f) \begin{align*}- \frac{5}{7} g > - 5\end{align*}
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