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Name: Date: Mathematics 1111 Exam 1 1 2 3 4 5 6 Score (1) (20 points) Determine whether the following statements are true or false. If the statement is true, explain why the given statement is true (a proof is not necessary), otherwise provide a counterexample. (a) For real numbers a, b, and c with a 6= 0 the quadratic equation ax2 + bx + c = 0 either has exactly two real solutions or no solutions for x. (b) For a positive real number c the interval satisfying the inequality |x| > c is given by (−c, c). 2 (2) (50 points) Solve for x in the following equations. Give exact solutions for each problem. (a) 9 − |4x + 7| = 5 (b) 2 5 4 − = 2 x−1 x+1 x −1 3 (c) x2 + 4x − 10 = 0 (d) x3 − 3x2 − 3x + 9 = 0 4 (e) x− √ x+1= x 3 5 (3) (10 points) A plumber and his assistant work together to replace the pipes in an old house. The plumber charges $50 an hour for his own labor and $20 an hour for his assistant’s labor. The plumber works twice as long as his assistant on this job, and the labor charge on the final bill is $5400. How long did the plumber and his assistant work on this job? 6 (4) (10 points) Phyllis invests $12, 000, a portion earning a simple interest rate of 40% per year and the rest earning a rate of 10% per year. After 1 year the total interest earned on these investments was $4200. How much money did she invest at each rate? 7 (5) (10 points) Kiran ran from Tortula to Cactus, a distance of 16 miles. She increased her speed by 2 miles per hour for the 6 mile trip from Cactus to Dry Junction. If the total trip took 5 hours, then what was her speed from Tortula to Cactus? 8 (6) (10 points) The equation 2x + kx − 7 = kx − k + 1 is really a family of equations, because for each value of k, we get a different equation for the unknown x. The letter k is called a parameter for this family. What value should we pick for k to make x = 3 a solution for the resulting equation.