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4.6

Situations That Always or Never Happen. 4.6. Comparing Situations Algebraically. Job A = 30,000 + 5000y Job B = 28,000 + 5000y When is the pay in Job A better than the pay in Job B? Write an equation or inequality to represent the situation. Solve. Sentences that are Always True.

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4.6

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  1. Situations That Always or Never Happen 4.6

  2. Comparing Situations Algebraically • Job A = 30,000 + 5000y Job B = 28,000 + 5000y • When is the pay in Job A better than the pay in Job B? • Write an equation or inequality to represent the situation. • Solve.

  3. Sentences that are Always True • Since 30,000 > 28,000 is always true, the disappearance of y signals that y can be any real number. Job A will always pay better than Job B.

  4. Comparing Situations Algebraically (continued) • Job A = 30,000 + 5000y Job B = 28,000 + 5000y • When is the pay in Job A less than the pay in Job B? • Write an equation or inequality to represent the situation. • Solve.

  5. Sentences that are Never True • Since 30,000 < 28,000 is never true, the disappearance of y signals that y can be any real number. Job A will never pay less than Job B.

  6. Determining True or False Sentences • If you are solving and the variable disappears…

  7. Steps to determining Situations that Always or Never Happen • Follow the steps to solve ax + b = cx + d or ax + b < cx + d. • Look at end result to determine if the resulting sentence is true (will always happen  all real solutions) or false (will never happen  no real solution) • Check by looking at a graph.

  8. Check with a Graph • Using your graphing calculator, set Y1 = left side of equation or inequality and Y2 = right side of equation or inequality • for equations (ax + b = cx + d) • If lines are parallel (never cross)  no real solution • If lines overlap (looks like one line)  all real solutions • If lines cross  x value of intersection point is solution • for inequalities (ax + b < cx + d) • If lines are parallel, line to the left > line to right, line to right < line to left • Use above information to compare to original statement to determine if true or false. • If lines overlap (looks like one line)  no real solutions (if < or >), all real numbers (if ≤ or ≥) • If lines cross  x value of intersection point is solution

  9. Example 1 • Solve 18m – 20 + 3m = 3(7m + 2) • Follow the steps to solving ax + b = cx + d • Look at the result and determine if it is a true or false statement. • Check with a graph.

  10. Example 2 • Solve 42k < 80k + 6 – 38k • Follow the steps to solving ax + b < cx + d • Look at the result and determine if it is a true or false statement. • Check with a graph.

  11. Example 3 – you try! • Solve 7x – 10 – 3x = 2(2x – 4)

  12. Review • Hamburger Heaven pays a starting salary of $6.90 an hour, and each year increases it by $1 an hour. Video King starts at $7.50 an hour and also increases $1 an hour per year. • Write an expression representing the salary of Hamburger Heaven and one representing the salary of Video King. • Use algebra to determine when does Hamburger Heaven pay more? • Solve 40 – 2x = 11 + 29 – 2x

  13. Review Answers • Hamburger Heaven pays a starting salary of $6.90 an hour, and each year increases it by $1 an hour. Video King starts at $7.50 an hour and also increases $1 an hour per year. • H = 6.90 + y V = 7.50 + y • 6.90 + y > 7.50 + y  6.90 > 7.50  no solution so, Hamburger Heaven never pays more. • 40 = 40  true  all real solutions

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