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7.1: Linear Inequalities

Difficulty Level: At Grade Created by: CK-12

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: \begin{align*}x \ge 4\end{align*}. Press PRGM to access the Program menu and choose the LINEQUA program.

Enter the left side of the inequality, \begin{align*}X\end{align*}, and press ENTER.

Then enter the right side of the inequality, \begin{align*}4\end{align*}, and press ENTER.

Select an inequality symbol. Press \begin{align*}2\end{align*} to choose \begin{align*}\ge\end{align*}.

Choose 1:View Table to see a table of values.

The calculator displays a table with several columns. The first column \begin{align*}X\end{align*} shows the values of the variable, \begin{align*}x\end{align*}. The second column \begin{align*}Y1\end{align*} shows the value of the left side for each \begin{align*}x-\end{align*}value. The third column \begin{align*}Y2\end{align*} shows the value of the right side for each \begin{align*}x-\end{align*}value.

1. Describe the numbers in the \begin{align*}Y1\end{align*} column. How do they compare to the \begin{align*}x-\end{align*}values? Explain.

2. Describe the numbers in the \begin{align*}Y2\end{align*} column. Are they affected by the \begin{align*}x-\end{align*}values? Explain.

Now look at the fourth column \begin{align*}Y3\end{align*}. Each entry in this column is either a \begin{align*}1\end{align*} or a \begin{align*}0\end{align*}. Examine this column.

(Note: To scroll up or down, return to the \begin{align*}X\end{align*} column, scroll, and then return to the \begin{align*}Y3\end{align*} column.)

3. For what \begin{align*}x-\end{align*}values is there a \begin{align*}1\end{align*} in the \begin{align*}Y3\end{align*} column?

4. Substitute one of these \begin{align*}x-\end{align*}values into \begin{align*}x \ge 4\end{align*}. Is the inequality true for this value of \begin{align*}x\end{align*}?

5. For what \begin{align*}x-\end{align*}values is there a \begin{align*}0\end{align*} in the \begin{align*}Y3\end{align*} column?

6. Substitute one of these \begin{align*}x-\end{align*}values into \begin{align*}x \ge 4\end{align*}. Is the inequality true for this value of \begin{align*}x\end{align*}?

Press ENTER to exit the table, then press ENTER again to select 1:Another Ineq.

This time enter the inequality \begin{align*}x < -2\end{align*}.

Choose 1:View Table from the menu. Look at each column. Again, each entry in the \begin{align*}Y3\end{align*} column is either a \begin{align*}1\end{align*} or a \begin{align*}0\end{align*}.

7. For what \begin{align*}x-\end{align*}values is there a \begin{align*}1\end{align*} in the \begin{align*}Y3\end{align*} column?

8. Substitute one of these values in for \begin{align*}x\end{align*} in \begin{align*}x < -2\end{align*}. Is the inequality true for this value of \begin{align*}x\end{align*}?

9. For what \begin{align*}x-\end{align*}values is there a \begin{align*}0\end{align*} in the \begin{align*}Y3\end{align*} column?

10. Substitute one of these values in for \begin{align*}x\end{align*} in \begin{align*}x < -2\end{align*}. Is the inequality true for this value of \begin{align*}x\end{align*}?

11. Complete each statement.

a) If the \begin{align*}x-\end{align*}value makes the inequality true, the entry in the \begin{align*}Y3\end{align*} column is _______.

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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