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13.5 Solve Two-Step Equations and Inequalities with Fractions Common Core Standards 7.EE.4. Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width? b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solution. WARM-UP Solve the two-step equations and inequalities. 1) 3x − 7 > −40 2) −17 = Find the reciprocal. 3) 5 4) 1 3 6 x − 10 6 Solve Two-Step Equations and Inequalities with Fraction Coefficients Can we use the Inverse Property of Multiplication to help us solve equations? 1 7 g =1 7 1 NOTES To remove a fraction coefficient multiply both sides by the reciprocal during step 2. Step 1 remains the same. Examples Solve the equations and inequalities. 1 x + 6 < 16 2 1 x−3= 4 5 EXAMPLES Solve the equations and inequalities. 2 x−8≥2 5 9 x + 12 = 21 2 NOTES The formula to convert Fahrenheit temperatures into Celsius is… 9 C + 32 = F 5 Example The temperature is 50o Fahrenheit. What is the temperature in Celsius? EXAMPLES Use the formula to convert 68o Fahrenheit into Celsius. 9 C + 32 = F 5 NOTES It doesn’t matter where the negative sign is in the fraction. Examples Solve the equations. 1 − x+5=7 4 −2 2= x−6 7 PRACTICE Solve the equations and inequalities. 1 x−6>5 3 2 −8 = x − 2 5 Use the formula to convert 104o Fahrenheit into Celsius. 9 C + 32 = F 5 PRACTICE A sunflower grows a rate of ¼ foot per week. When it was planted in the ground it was 2 feet tall. If the sunflower is currently 5 feet tall, write and solve a twostep equation to determine the age in weeks (w) of the flower. FINAL QUESTION Solve the inequality (hint: do we need to flip the symbol?) 5 − x+7<2 8