Presentation on theme: "Solving Inequalities by Adding or Subtracting"— Presentation transcript:
1Solving Inequalities by Adding or Subtracting 3-2Solving Inequalities byAdding or SubtractingHolt Algebra 1Warm UpLesson PresentationLesson Quiz
2Warm Up Graph each inequality. Write an inequality for each situation. 1. The temperature must be at least –10°F.2. The temperature must be no more than 90°F.–1010x ≥ –10x ≤ 90–9090Solve each equation.3. x – 4 = 10144. 15 = x + 1.113.9
3Objectives Solve one-step inequalities by using addition. Solve one-step inequalities by using subtraction.
4Solving one-step inequalities is much like solving one-step equations Solving one-step inequalities is much like solving one-step equations. To solve an inequality, you need to isolate the variable using the properties of inequality and inverse operations.
6Helpful HintUse an inverse operation to “undo” the operation in an inequality. If the inequality contains addition, use subtraction to undo the addition.
7Example 1A: Using Addition and Subtraction to Solve InequalitiesSolve the inequality and graph the solutions.x + 12 < 20x + 12 < 20Since 12 is added to x, subtract 12 from both sides to undo the addition.–12 –12x + 0 < 8x < 8Draw an empty circle at 8.–10–8–6–4–2246810Shade all numbers less than 8 and draw an arrow pointing to the left.
8Example 1B: Using Addition and Subtraction to Solve InequalitiesSolve the inequality and graph the solutions.d – 5 > –7d + 0 > –2d > –2d – 5 > –7Since 5 is subtracted from d, add 5 to both sides to undo the subtraction.Draw an empty circle at –2.–10–8–6–4–2246810Shade all numbers greater than –2 and draw an arrow pointing to the right.
9Example 1C: Using Addition and Subtraction to Solve InequalitiesSolve the inequality and graph the solutions.0.9 ≥ n – 0.30.9 ≥ n – 0.3Since 0.3 is subtracted from n, add 0.3 to both sides to undo the subtraction.1.2 ≥ n – 01.2 ≥ n1.2Draw a solid circle at 1.2.12Shade all numbers less than 1.2 and draw an arrow pointing to the left.
10Solve each inequality and graph the solutions. Check It Out! Example 1Solve each inequality and graph the solutions.a. s + 1 ≤ 10Since 1 is added to s, subtract 1 from both sides to undo the addition.s + 1 ≤ 10–1 –19–10–8–6–4–2246810s + 0 ≤ 9s ≤ 9b > –3 + tSince –3 is added to t, add 3 to both sides to undo the addition.> –3 + t+3> 0 + t–10–8–6–4–2246810t <
11Solve the inequality and graph the solutions. Check It Out! Example 1cSolve the inequality and graph the solutions.q – 3.5 < 7.5Since 3.5 is subtracted from q, add 3.5 to both sides to undo the subtraction.q – 3.5 < 7.5q – 0 < 11q < 11–7–5–3–1135791113
12Since there can be an infinite number of solutions to an inequality, it is not possible to check all the solutions. You can check the endpoint and the direction of the inequality symbol.The solutions of x + 9 < 15 are given by x < 6.
13Example 2: Problem-Solving Application Sami has a gift card. She has already used $14 of the of the total value, which was $30. Write, solve, and graph an inequality to show how much more she can spend.Understand the problem1The answer will be an inequality and a graph that show all the possible amounts of money that Sami can spend.List important information:• Sami can spend up to, or at most $30.• Sami has already spent $14.
14Example 2 Continued2Make a PlanWrite an inequality.Let g represent the remaining amount of money Sami can spend.Amount remainingplus$30.is at mostamount usedg+14≤30g + 14 ≤ 30
15Draw a solid circle at 0 and16. Example 2 ContinuedSolve3g + 14 ≤ 30Since 14 is added to g, subtract 14 from both sides to undo the addition.– 14 – 14g + 0 ≤ 16g ≤ 16Draw a solid circle at 0 and16.24681012141618Shade all numbers greater than 0 and less than 16.
16 Example 2 Continued Look Back 4 Check Check a number less than 16. g ≤ 30≤ 3020 ≤ 30Check the endpoint, 16.g + 14 = 3030 30Sami can spend from $0 to $16.
17Check It Out! Example 2The Recommended Daily Allowance (RDA) of iron for a female in Sarah’s age group (14-18 years) is 15 mg per day. Sarah has consumed 11 mg of iron today. Write and solve an inequality to show how many more milligrams of iron Sarah can consume without exceeding RDA.
18Check It Out! Example 2 Continued Understand the problem1The answer will be an inequality and a graph that show all the possible amounts of iron that Sami can consume to reach the RDA.List important information:• The RDA of iron for Sarah is 15 mg.• So far today she has consumed 11 mg.
19Check It Out! Example 2 Continued Make a PlanWrite an inequality.Let x represent the amount of iron Sarah needs to consume.Amount takenplus15 mgis at mostamount needed11+x1511 + x 15
20Check It Out! Example 2 Continued Solve311 + x 15Since 11 is added to x, subtract 11 from both sides to undo the addition.– –11x 4Draw a solid circle at 4.12345678910Shade all numbers less than 4.x 4. Sarah can consume 4 mg or less of iron without exceeding the RDA.
21Check It Out! Example 2 Continued Look Back4CheckCheck a number less than 4. 15 1514 15Check the endpoint, 4.11 + x = 15Sarah can consume 4 mg or less of iron without exceeding the RDA.
22Example 3: ApplicationMrs. Lawrence wants to buy an antique bracelet at an auction. She is willing to bid no more than $550. So far, the highest bid is $475. Write and solve an inequality to determine the amount Mrs. Lawrence can add to the bid. Check your answer.Let x represent the amount Mrs. Lawrence can add to the bid.$475plusamount can addis at most$550.x+475≤550475 + x ≤ 550
23Example 3 Continued475 + x ≤ 550– – 475x ≤ 750 + x ≤ 75Since 475 is added to x, subtract 475 from both sides to undo the addition.Check the endpoint, 75.Check a number less than 75.x = 550x ≤ 550≤ 550525 ≤ 550Mrs. Lawrence is willing to add $75 or less to the bid.
24Let p represent the number of additional pounds Josh needs to lift. Check It Out! Example 3What if…? Josh wants to try to break the school bench press record of 282 pounds. He currently can bench press 250 pounds. Write and solve an inequality to determine how many more pounds Josh needs to lift to break the school record. Check your answer.Let p represent the number of additional pounds Josh needs to lift.250 poundsplusadditional poundsis greater than282 pounds.250+p>282
25Check It Out! Example 3 Continued – –250p > 32Since 250 is added to p, subtract 250 from both sides to undo the addition.CheckCheck the endpoint, 32.Check a number greater than 32.p = 282p > 282> 282283 > 282Josh must lift more than 32 additional pounds to reach his goal.
26Lesson Quiz: Part ISolve each inequality and graph the solutions.1. 13 < x + 7x > 62. –6 + h ≥ 15h ≥ 21y ≤ –2.1y ≤ –8.8
27Lesson Quiz: Part II4. A certain restaurant has room for 120 customers. On one night, there are 72 customers dining. Write and solve an inequality to show how many more people can eat at the restaurant.x + 72 ≤ 120; x ≤ 48, where x is a natural number