1. Inequalities Introduction Students will identify and draw graphs of inequalities. x < -4 a < 9 n > 9 y < 4 x > 32 m > -40 k < -8 n > -26

2. Inequality Symbols Inequalities use symbols to compare numbers and variables: > is the “greater than” symbol < is the “less than” symbol > is the “greater than or equal to” symbol ( The line under the symbol means “or equal”.) < is the “less than or equal to” symbol (The line under the symbol means “or equal”.)

3. Understand the Symbols Compare the trees using a greater than or less than symbol. The first tree is smaller (or less) than the second so we use the “less than” symbol.

4. Understand the Symbols Compare the hearts using a greater than or less than symbol. The first heart is bigger (or greater) than the second so we use the “greater than” symbol.

5. Example 1. Identify the Graph -2 -1 0 1 2 3 4 Notice that the “dot” at the beginning of the dark shaded ray is NOT shaded in. That means that the variable (letter) is NOT equal to that number. The graph is shaded darker to the right which means the graph is greater than 1. We write the inequality: x > 1

6. Example 2. Identify the Graph -2 -1 0 1 2 3 4 Notice that the “dot” at the beginning of the dark shaded ray is NOT shaded in. That means that the variable (letter) is NOT equal to that number. The graph is shaded darker to the left which means the graph is less than 3. We write the inequality: x < 3

7. Example 3. Identify the Graph -6 -5 -4 -3 -2 -1 0 Notice that the “dot” at the beginning of the ray is shaded. That means that the variable (letter) is equal to that number. The graph is shaded darker to the right which means the graph is greater than or equal to -3. We write the inequality: x > -3 Remember, the line under the symbol means that the variable also equals the number.

8. Example 4. Identify the Graph -9 -8 -7 -6 -5 -4 -3 Notice that the “dot” at the beginning of the ray is shaded. That means that the variable (letter) is equal to that number. The graph is shaded darker to the left which means the graph is less than or equal to -6. We write the inequality: x < -6 Remember, the line under the symbol means that the variable also equals the number.

9. Handy hint to remember: When graphing an inequality, always put the variable first. Example: n < -13 Then, the side of the “dot” that is shaded can be found by thinking of the symbol as an arrowhead. Since the symbol is underlined, draw a solid dot. The < symbol looks like an arrow pointing to the left. That means we shade to the left.

10. Practice: Identify the graph -2 -1 0 1 2 3 4 10 11 12 13 14 15 16 -9 -8 -7 -6 -5 -4 -3 18 20 22 24 -6 -4 -2 0 x < 1 x > 11 x < -5 x > 21 x < -3 “x is less than or equal to 1” “x is greater than 11” “x is less than -5” “x is greater than 21” “x is less than or equal to -3”

11. Example 1: Draw the graph. 3 4 5 6 7 8 9 x > 5 On the graph find the point 5. Since the symbol is not underlined… draw an open dot (not shaded in) at 5. Since the > symbol looks like an arrowhead pointing to the right… shade to the right on the graph.

12. Example 2: Draw the graph. 3 4 5 6 7 8 9 x < 7 On the graph find the point 7. Since the symbol is not underlined… draw an open dot (not shaded in) at 7. Since the < symbol looks like an arrowhead pointing to the left… shade to the left on the graph.

13. Example 3: Draw the graph. -8 -6 -4 -2 x > -7 On the graph find the point -7. Since the symbol is underlined… draw a shaded dot at -7. Since the > symbol looks like an arrowhead pointing to the right… shade to the right on the graph.

14. Example 4: Draw the graph. -40 -30 -20 -10 x < -25 On the graph find the point -25. Since the symbol is underlined… draw a shaded dot at -25. Since the < symbol looks like an arrowhead pointing to the left… shade to the left on the graph.

15. Practice: Draw the graph. x < 9 5 6 7 8 9 10 11 x > -5 -8 -6 -4 -2 x < 6 2 4 6 8 x > -75 -100 -50 0 x < 11 6 7 8 9 10 11 12

16. Important Note: Sometimes an inequality is written backwards… with the number first. Example: -34 > x Re-write the inequality with the variable FIRST. x -34 BUT, you must switch the symbol so it points the opposite direction. <

17. Examples: Re-write the Inequalities. 4 > x is the same as x < 4 -9 < x is the same as x > -9 11 < x is the same as x > 11 -24 > x is the same as x < -24

18. Summary -2 -1 0 1 2 3 4 Open Dot means it IS NOT equal to the number where it is located. Use > or < symbol. -9 -8 -7 -6 -5 -4 -3 A shaded Dot means it IS equal to the number where it is located. Use > or < symbol. The > symbol means shade to the right… like an arrow pointing right. Write the Inequality with the variable FIRST. The < symbol means shade to the left… like an arrow pointing left. STOP