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1.4

1.4. Equations and Inequalities. 1. 2. GOAL. GOAL. Check solutions and solve equations using mental math. Check solutions of inequalities in a real-life problems, such as regulating your cat’s caloric intake.

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1.4

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  1. 1.4 Equations and Inequalities 1 2 GOAL GOAL Check solutions and solve equations using mental math. Check solutions of inequalities in a real-life problems, such as regulating your cat’s caloric intake. To solve a real-life problem such as how long you must save money to buy a violin. Whatyou should learn Why you should learn it

  2. 1.4 Equations and Inequalities CHECKING AND SOLVING EQUATIONS 1 GOAL • VOCABULARY • equation— • open sentence— • solution of an equation— formed when an equal sign is placed between two expressions an equation that contains at least one variable a number which will make a true statement when substituted for the variable in a single-variable equation

  3. EXAMPLE 2 EXAMPLE 1 To check if a number is a possible solution to a variable equation, simply substitute the number for the variable. If the statement is true, the number is a solution. If the statement is false, the number is not a solution. Checking a possible solution

  4. Extra Example 1 Check whether the numbers 3 and 4 are solutions of the equation 5x – 7 = 8. x = 3: 5x – 7 = 8 x = 4: 5x – 7 = 8 5(4) – 7 = 8 5(3) – 7 = 8 8 = 8 13 ≠ 8 3 is a solution. 4 is not a solution.

  5. Check whether the numbers 5, 10, and 15 are solutions of the equation 12s + 5 = 125. x = 15: 12s + 5 = 125 x = 5: 12s + 5 = 125 x = 10: 12s + 5 = 125 Extra Example 2 12(5) + 5 = 125 12(10) + 5 = 125 12(15) + 5 = 125 65 ≠ 125 125 = 125 185 ≠ 125 Only 10 is a solution to the equation.

  6. ACTIVITY Using Mental Math to Solve Equations Developing Concepts EXAMPLE 3 • VOCABULARY • solving an equation— finding all the solutions of an equation For now, we will solve equations using mental math. Please work through the Activity on page 25.

  7. Extra Example 3 You need to buy ingredients to make a pizza. Cheese costs $5.99, pizza dough costs $2.39, tomato sauce costs $2.49, and pepperoni costs $2.98. You have a twenty-dollar bill. About how much change will you receive? Work this out in the best calculator you have—your brain! Hint: Round each amount to the nearest half-dollar. Possible solution: Rounding each amount gives $6 + $2.50 + $2.50 + $3 = $14. Subtracting gives $20 – $14 = $6. You will receive about $6 in change.

  8. Checkpoint • Check whether the numbers 3, 6, and 9 are solutions of the equations 8s – 20 = 52. • 2. Use mental math to solve:You need to buy supplies for school. A box of pencils costs $2.98, a package of pens costs $3.95, a notebook costs $4.49, and a ruler costs $0.89. You have $10. How much more money do you need? 9 is a solution; 3 and 6 are not. You need about $2.50.

  9. 1.4 Equations and Inequalities 2 CHECKING SOLUTIONS OF INEQUALITIES GOAL • VOCABULARY • inequality— • solution of an inequality— formed when an inequality symbol is placed between two expressions a number which will make a true statement when substituted for the variable in an inequality

  10. EXAMPLE 4 EXAMPLE 5 Inequality Symbols Do you know how to read each of the four inequality symbols? Let’s find out! Match each symbol with its meaning. < is greater than or equal to > is less than ≤ is less than or equal to ≥ is greater than Click to see the answers. To check if a number is a solution of an inequality, simply substitute the number for the variable. If it makes a true statement, the number is a solution.

  11. Extra Example 4 • Decide whether 3 is a solution of the inequality. • 3x – 2 > 5 b. x + 5 < 8 • c. x – 1 ≥ 2 d. 4x2 ≤ 32 YES NO YES NO

  12. 240t≥ 500 • 240(2) ≥ 500 • 480 ≥500 NO Extra Example 5 If you weigh 120 lbs, the number of calories you will burn while walking briskly is about 240 times the number of hours you walk. Using the following values of x, will a 120 lb person walking for x hours burn at least 500 calories? a. 2 hours b. 3 hours Hint: First, write a verbal model, then the inequality. Model: Calories burned ≥ 500 Labels/Inequality: 240 • number of hours≥ 500 240 • t ≥ 500 Solution: b. 240t≥ 500 240(3) ≥ 500 720 ≥500 YES

  13. Checkpoint • Decide whether 4 is a solution of the inequality. • a. 2x – 1 > 5 b. x – 2 > 6 • c. x + 3 ≥ 5 d. 3x2 ≤ 52 • Your scores on the last three tests were 82, 87, and 74. Your score on the next test is x points. Do the following values for x give you at least 320 total points? • a. 75 b. 81 NO YES YES YES NO YES

  14. QUESTIONS?

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