Preview of Grade 7 AF4.1

1. California Standards Preview of Grade 7 AF4.1 Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results.

2. When you solve two-step equations, you can use the order of operations in reverse to isolate the variable. You can use the same process when solving two-step inequalities.

3. Remember! Draw a closed circle when the inequality includes the point and an open circle when it does not include the point.

4. y 2 – 6 > 1 y 2 > 7 y 2 > (2)7 (2) – – – 21 14 7 0 7 14 21 Additional Example 1A: Solving Two-Step Inequalities Solve. Then graph the solution set on a number line. y 2 – 6 > 1 + 6 + 6 Add 6 to both sides. Multiply both sides by 2. y > 14

5. ? 4 > 1 - 6 > 1 Additional Example 1A Continued Check y - 6 > 1 2 20 20 is greater than 14 Substitute 20 for y. 2

6. m –3 5 ≥ + 8 m –3 -3 ≥ –3 0 3 6 9 12 15 m –3 (–3) (–3) ≤ (–3) Additional Example 1B: Solving Two-Step Inequalities Solve. Then graph the solution set on a number line. m –3 5 ≥ + 8 – 8 –8 Subtract 8 from both sides. Multiply both sides by –3, and reverse the inequality symbol. m ≥ 9 •

7. Additional Example 1C: Solving Two-Step Inequalities Solve. Then graph the solution on a number line. 4y – 5 < 11 4y – 5 < 11 + 5 + 5 Add 5 to both sides. 4y < 16 4y < 16 Divide both sides by 4. 4 4 y < 4 º 2 4 6 0 –6 –4 –2

8. 3 6 9 0 –9 –6 –3 Additional Example 1D: Solving Two-Step Inequalities Solve. Then graph the solution set on a number line. –4 ≥ –3x + 5 –4 ≥ –3x + 5 – 5 –5 Subtract 5 from both sides. –9 ≤ –3x Divide both sides by –3, and reverse the inequality symbol. –3 –3 3≤ x

9. h 7 + 1 > –1 h 7 > –2 h 7 > (7)(–2) (7) – – – 21 14 7 0 7 14 21 Check It Out! Example 1A Solve. Then graph the solution set on a number line. h 7 + 1 > –1 – 1 – 1 Subtract 1 from both sides. Multiply both sides by 7. h > –14

10. ? 2 > 1 + 1 > -1 Check It Out! Example 1A Continued Check h + 1 > -1 7 7 7 is greater than -14 Substitute 7 for h. 7

11. m –2 + 1 ≥ 7 m –2 ≥ 6 m –2 ≤ (6) (–2) (–2) –18 –12 –6 0 6 12 18 Check It Out! Example 1B Solve. Then graph the solution set on a number line. m –2 + 1 ≥ 7 – 1 –1 Subtract 1 from both sides. Multiply both sides by –2, and reverse the inequality symbol. • m ≤–12

12. Check It Out! Example 1C Solve. Then graph the solution on a number line. 2y – 4 > –12 2y – 4 > –12 + 4 + 4 Add 4 to both sides. 2y > –8 2y > –8 Divide both sides by 2. 2 2 y > –4 º 2 4 6 0 –6 –4 –2

13. 3 6 9 0 –9 –6 –3 Check It Out! Example 1D Solve. Then graph the solution set on a number line. –9x + 4 ≤ 31 –9x + 4 ≤ 31 – 4 –4 Subtract 4 from both sides. –9x ≤ 27 Divide both sides by –9, and reverse the inequality symbol. –9x ≥ 27 –9 –9 x ≥ –3

14. 12 252 , or 12 r ≤ Additional Example 2: Application Sun-Li has $30 to spend at the carnival. Admission is $5, and each ride costs $2. What is the greatest number of rides she can ride? Let r represent the number of rides Sun-Li can ride. 5 + 2r ≤ 30 – 5 –5 Subtract 5 from both sides. 2r ≤ 25 Divide both sides by 2. 2r ≤ 25 2 2 Sun-Li can ride only a whole number of rides, so the most she can ride is 12.

15. Check It Out! Example 2 Brice has $30 to take his brother and his friends to the movies. If each ticket costs $4.00, and he must buy tickets for himself and his brother, what is the greatest number of friends he can invite? Let t represent the number of tickets. 8 + 4t ≤ 30 – 8 –8 Subtract 8 from both sides. 4t≤ 22 4t≤ 22 Divide both sides by 4. 4 4 t ≤ 5.5 Brice can only buy a whole number of tickets, so the most people he can invite is 5.