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Algebra Chapter 2 Solving Linear Equations 2.1 Find Square Roots and Compare Real Numbers Square Root: __________________________________________________________________________ Radicand: ____________________________________________________________________________ Perfect Square: ________________________________________________________________________ Irrational Number: _____________________________________________________________________ Real Numbers: ________________________________________________________________________ Example 1: Find the square roots. 16 400 Example 2: Approximate 81 52 to the nearest integer. Example 3: Tell whether each of the following numbers is a real number, a rational number, an irrational number, an integer, or a whole number: 64 17 36 Number Real number? Rational number? Irrational number? Integer? Whole number? 64 17 36 Example 4: Order the numbers from least to greatest: 3 , 16 ,2.2, 12 , 6 5 Example 5: Rewrite the given conditional statement in if-then form. Then tell whether the statement is true or false. If false, give a counterexample. a. Given: No square roots are rational numbers. b. Given: All integers are rational numbers. 2.2 Solve One-Step Equations Inverse operations: _____________________________________________________________________ Equivalent equations: ___________________________________________________________________ Reciprocal: ___________________________________________________________________________ Properties of Equality Addition Property of Equality: ____________________________________________________________ Subtraction Property of Equality: _________________________________________________________ Multiplication Property of Equality: _______________________________________________________ Division Property of Equality: ____________________________________________________________ Example 1: Solve x + 11 = 15. Example 2: Solve x – 8 = 17. Example 3: Solve 7x = –63. Example 4: Solve Example 5: Solve x =4 12 3 x6 5 Example 6: In the 2004 Summer Olympics, Inge de Bruijn won the women’s 50-meter freestyle. Her winning time was 24.58 seconds. Find her average swimming speed to the nearest hundredth of a meter per second. 2.3 Solve Two-Step Equations Example 1: Solve 3x – 7 = 8. Example 2: Solve 11x – 9x = 14. Example 3: The output of a function is 7 more than twice the input. Find the input when the output is 13. Example 4: To rent a booth at the county fairgrounds cost $42 per day plus a onetime equipment fee of $85. Find the number of days Mr. Batzle rented a booth if he paid a total of $337. Write and solve an equation to find the answer. 2.4 Solve Multi-Step Equations Example 1: Solve 17x – 11x + 8 = 20. Example 4: Solve Example 2: Solve 4x + 3(2x – 1) = 17. 3 (5x – 4) = 12. 4 Example 5: Edmund and Roberto took a 7 day (168 hours), 90 mile canoe trip down the Allagash River. If they paddled at an average rate of 2.5 miles per hour, how many hours did they not spend paddling? Write an equation to find the answer. 2.5 Solve Equations with Variables on Both Sides Identity: _____________________________________________________________________________ Example 1: Solve 13 – 6x = 3x – 14. Example 2: Solve 4x – 7 = 3 (9x – 15) 4 Example 3: A music website sold 94 single songs and 67 albums today. The number of single downloads has been increasing by 22 each day. The number of album downloads has been decreasing by 5 each day. If these trends continue, in how many days will the number of single downloads will be ten times the number of album downloads? Write and solve an equation to find the number of days. Example 4: Solve the equation, if possible. a. 4(3x – 2) = 2(6x + 1) b. 4(4x – 5) = 2(8x – 10) 2.6 Write Ratios and Proportions Ratio: _______________________________________________________________________________ Proportion: ___________________________________________________________________________ Simplest form: ________________________________________________________________________ Example 1: A shoe store sells 15 pairs of women’s shoes and 12 pairs of men’s shoes. Find the specified ratio. a. Find the ratio of women’s shoe sales to men’s shoe sales. b. Find the ratio of men’s shoe sales to total shoe sales. Example 2: Solve a proportion 7 9 = x 54 Example 3: A restaurant owner uses 3 cloves of garlic for every 5 pints of sauce. The restaurant uses 210 pints of sauce during the day. Find the number of cloves of garlic the restaurant uses to make the sauce. 2.7 Solve Proportions Using Cross Products Cross Product: ________________________________________________________________________ Scale Drawing: ________________________________________________________________________ Scale Model: _________________________________________________________________________ Scale: _______________________________________________________________________________ 20 8 Example 1: Solve = 35 x Example 3: A bag of large breed dog food recommends feeding a dog 3 cups of food a day for every 40 pounds of body weight. A dog weights 98 pounds. How much food should the dog be eating each day? Example 4: A blueprint of an office building has a scale of 2 inches:15 feet. A completed scale model of the building is about 14.5 inches tall. Estimate the actual height of the office building. 2.8 Rewrite Equations and Formulas Literal Equation: ______________________________________________________________________ Formula: _____________________________________________________________________________ Example 1: Solve p + qx = r for x. Then use the solution to solve 3 + 5x = -7. Example 2: Write 9x – 4y = 8 so that y is a function of x. Example 3: The formula for the volume of a rectangular prism is V = lwh. Solve the formula for l. Example 4: The area A of a triangle is given by the formula A = 1 bh where b is the base and h is the height. 2 a. Solve the formula for the base b. b. Use the rewritten formula to find the base of the triangle shown, which has an area of 106.8 square inches. Example 5: Irina deposited $650 in a savings account. After two years her account balance was $682.50. Find the rate of interest for the two years. Use the formulas A =P(1 +rt), where A is the account balance, P is the principal, r is the rate, and t is the time. Rewrite the formula to isolate r and then solve.