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Solving Equations with Decimals: Exact and Approximate Solutions

Learn how to use the order of operations and rounding to find the exact and approximate solutions of equations that contain decimals. Explore scenarios involving practical rounding for cost-sharing and changing decimal coefficients to integers for easier calculation.

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Solving Equations with Decimals: Exact and Approximate Solutions

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  1. OBJECTIVE • I will use the order of operations and rounding to find the exact and approximate solutions of equations that contain decimals.

  2. Rounding for a Practical Answer Six people buy some food to take to the lake. The total cost for the food is $33.45. If each person pays the same amount, how much does each person owe? 6x = 33.45 Original equation x = 5.575 Exact answer x ≈ 5.58 Round to the nearest cent Each person owes $5.58. The exact answer is not practical because you cannot have a fraction of a cent, but six times the rounded answer does not correspond to the cost of the food. It is one cent too much due to round-off error.

  3. Rounding for the Final Answer Solve 7x + 32 = 128. Round to the nearest hundredth. 7x + 32 = 128 Original equation 7x = 96 Subtract 32 from each side x = 13.714… Divide both sides by 7 x ≈ 13.71 Round to nearest hundredth The solution is approximately 13.71.

  4. Guided Practice 1. Solve -12x - 13 = 37. Round to the nearest hundredth.

  5. Original Equation Involving Decimals • Solve using same methods used to solve equations without a decimal. Solve 2.14x + 3.81 = 0.53x + 14.96. Round to the nearest hundredth. 2.14x + 3.81 = 0.53x + 14.96 1.61x = 11.15 x = 6.925… x ≈ 6.93

  6. Guided Practice 1. Solve 7.82x - 3.14 = 4.66x + 2.71. Round your answer to the nearest hundredth.

  7. Changing Decimal Coefficients to Integers • Change decimals to integers for easier calculation Solve 2.3x + 6.2 = 1.5x + 9.8. Round to the nearest tenth. 2.3x + 6.2 = 1.5x + 9.7 Original Equation 23x + 62 = 15x + 97 Multiply both sides by 10 8x = 35 Combine like terms x = 4.625 x ≈ 4.6

  8. Guided Practice Re-write the coefficients as integers and solve. Round to the nearest tenth. 1. 5.7 - 3.4x = 2.2x - 7.1

  9. Using a Verbal Model You are shopping for roses. The total tax is 10%. You have a total of $27.79 to spend. What is your price limit for the roses? Total cost is price limit and tax based on price limit.

  10. Using a Verbal Model Cost of Roses + Cost of Tax = Price Limit x + 0.10x = 27.79 Substitute 1.10x = 27.79 Combine like terms x = 25.263… Divide both sides by 1.10 x ≈ 25.26 Round down You have a limited amount of money to spend, so you must round down to $25.26. If you round up, the tax and total cost will increase and you will be a penny short.

  11. Guided Practice You are shopping for a bracelet. The sales tax is 5%. You have a total of $125.40 to spend. What is your price limit for the bracelet?

  12. Independent Practice Round answers to the nearest hundredth. • 3x + 5 = 20 • 2.51x - 4.27 = 1.13x + 2.15 • Change the decimal coefficients to integers to solve. 3.4x - 2.3 = 1.7x + 5.4

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