1 / 27

California Standards

California Standards. 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. Vocabulary. inequality

muhammad
Download Presentation

California Standards

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. California Standards 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. Vocabulary inequality algebraic inequality solution set

  3. An inequalitycompares two expressions using <, >, , or . is less than Fewer than, below is greater than More than, above is less than or equal to At most, no more than is greater than or equal to At least, no less than An inequality that contains a variable is an algebraic inequality.

  4. A solution of an inequality is any value of the variable that makes the inequality true. All of the solutions of an inequality are called the solution set. You can graph the solution set on a number line. The symbols < and > indicate an open circle.

  5. This open circle shows that 5 is not a solution. a > 5 The symbols ≤ and ≥ indicate a closed circle. This closed circle shows that 3 is a solution. b ≤ 3

  6. Draw a closed circle at –2 and all values of z greater than -2 . So shade to the right of –2 . 1 2 1 2 1 2 Additional Example 3: Graphing Inequalities Graph each inequality. A. –1> y Draw an open circle at –1. The solutions are all values of y less than –1, so shade the line to the left of –1. –3 –2 –1 0 1 2 3 12 B. z ≥ –2 –3 –2 –1 0 1 2 3

  7. Check It Out! Example 3 Graph each inequality. A. n < 3 Draw an open circle at 3. The solutions are all values of n less than 3, so shade the line to the left of 3. –3 –2 –1 0 1 2 3 B. a ≥ –4 Draw a closed circle at –4. The solutions are all values greater than –4, so shade to the right of –4. –6 –4 –2 0 2 4 6

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

  9. When you multiply (or divide) both sides of an inequality by a negative number, you must reverse the inequality symbol to make the statement true.

  10. 1 2 3 4 5 6 7 12 4x > 4 4 Example 1: Solving Two-Step Inequalities Solve and graph. 4x + 1 > 13 4x + 1 > 13 – 1– 1Since 1 is added to 4x, subtract 1 from both sides. 4x > 12 Since x is multiplied by 4, divide both sides by 4. x > 3

  11. Example 1 Continued Check According to the graph 4 should be a solution and 2 should not be a solution. x > 3 x > 3 Substitute 4 for x. Substitute 2 for x. ? ? 4 > 3 2 > 3  So 4 is a solution. So 2 is not a solution.

  12. -6 -5 -4 -3 -2 -1 0 18 –9x –9 –9 Example 2: Solving Two-Step Inequalities Solve and graph. –9x + 7  25 –9x + 7  25 – 7– 7Subtract 7 from both sides. –9x 18 Divide each side by –9; change  to . x–2

  13. 1 2 3 4 5 6 7 10 5x > 5 5 Check It Out! Example 3 Solve and graph. 5x + 2 > 12 5x + 2 > 12 – 2– 2Subtract 2 from both sides. 5x > 10 Divide both sides by 5. x > 2

  14. Check It Out! Example 3 Continued Check According to the graph 4 should be a solution and 1 should not be a solution. x > 2 x > 2 Substitute 4 for x. Substitute 1 for x. ? ? 4 > 2 1 > 2  So 4 is a solution. So 1 is not a solution.

  15. -6 -5 -4 -3 -2 -1 0 16 –4x –4 –4 Check It Out! Example 4 –4x + 2  18 –4x + 2  18 – 2– 2Subtract 2 from both sides. –4x 16 Divide each side by –4; change  to . x–4

  16. Example 5 < 10 + 4y 18 < Simplify. 4y 8 Divide each side by 4. 4y 8 < 4 4 < Simplify. y 2 0 1 2 3 4 5 Solving and Graphing a Two-Step Inequality Original inequality < 10 + 4y Subtract 10 from each side. 18 -10 -10

  17. Example 6 Multiple Choice Practice x 15 x 15 x 25 x 25 ≤ ≥ ≤ ≥ SOLUTION To find your profit, subtract the total costs from the total ticket sales. This amount should equal or exceed the minimum desired profit of $200. You are organizing a bowling night for charity. Each ticket costs $10 and includes shoe rental. Door prizes cost you $50. Which inequality describes the possible numbers x of people who need to attend for you to make a profit of at least $200?

  18. Example 6 Multiple Choice Practice – Write an inequality. 10x 50 200 ≥ 10x x 250 25 ≥ ≥ ANSWER The correct answer is D. +50 +50 Add 50 to each side. Divide each side by 10. 10 10

  19. Example 7 < – 4x Combine like terms. – – – – – – 4 4 4 4x 14 14 Divide each side by and reverse the inequality symbol. > 7 x > Simplify. 2 0 1 2 3 4 5 Combining Like Terms < x – – Original inequality 3x 14 < Subtract 3x from each side. 3x – 14 x – – 3x – 3x

  20. Guided Practice 1. – ANSWER 57 z ≤ 7z 15 – + ≥ 6 < 2. < ANSWER – n + 5 11n 40 3n Solve the inequality. Then graph the solution.

  21. Guided Practice Solve the inequality. Then graph the solution. 2 3. ANSWER – – y 16 9y 18 > > 9 for Examples 1, 2, and 3

  22. Guided Practice 4. WHAT IF? In Example 3, how many people need to attend for you to make a profit of at least $250? x ANSWER 30 people ≥ for Examples 1, 2, and 3

  23. Example 8: School Application A school’s Spanish club is selling bumper stickers. They bought 100 bumper stickers for $55, and they have to give the company 15 cents for every sticker sold. If they plan to sell each bumper sticker for $1.25, how many do they have to sell to make a profit? In order for the Spanish club to make a profit, the revenue must be greater than the cost. 1.25x > 55 + 0.15x

  24. 55 > 1.10x 1.10 1.10 Example 8 Continued 1.25x > 55 + 0.15x Subtract 0.15x from both sides. – 0.15x– 0.15x 1.10x > 55 Divide both sides by 1.10. x > 50 The Spanish club must sell more than 50 bumper stickers to make a profit.

  25. Check It Out! Example 9 A school’s French club is selling bumper stickers. They bought 200 bumper stickers for $45, and they have to give the company 25 cents for every sticker sold. If they plan to sell each bumper sticker for $2.50, how many do they have to sell to make a profit? In order for the French club to make a profit, the revenue must be greater than the cost. 2.5x > 45 + 0.25x

  26. 45 > 2.25x 2.25 2.25 Check It Out! Example 9 Continued 2.5x > 45 + 0.25x Subtract 0.25x from both sides. – 0.25x– 0.25x 2.25x > 45 Divide both sides by 2.25. x > 20 The French club must sell more than 20 bumper stickers to make a profit.

  27. 1 2 3 4 5 6 7 -10 -9 -8 -7 -6 -5 -4 -18 -17 -16 -15 -14 -13 -12 Lesson Quiz: Part I Solve and graph. 1. 4x – 6 > 10 2. 7x + 9 < 3x – 15 3.w – 3w < 32 x > 4 x < –6 w > –16

More Related