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Math 1010-2 Midterm 1 Solution Instructions: There are TEN problems in this exam. Answer the questions in the spaces provided on the question sheets. If you need more space, use the bottom of the last page, but remember to indicate which problem you are doing. Partial credit will be awarded. Calculators are NOT allowed. This exam is closed book and closed notes. Show you work on each problem. 1. (10 points) Suppose f (x) = (x + 1)2 + 2x − 1. Evaluate f (y + 1) − f (1) Solution: Simplify f (x), we have f (x) = (x2 + 2x + 1) + 2x − 1 = x2 + 4x So f (y + 1) = (y + 1)2 + 4(y + 1) = (y 2 + 2y + 1) + 4y + 4 = y 2 + 6y + 5 and f (1) = 12 + 4 × 1 = 5 Therefor f (y + 1) − f (1) = (y 2 + 6y + 5) − 5 = y 2 + 6y 2. (10 points) Simplify the following expression: −(a + 2)2 + (−a)[−(a − 2) + 2a] Solution: = = = = −(a + 2)2 + (−a)[−(a − 2) + 2a] −(a + 2)(a + 2) + (−a)[a + 2] −[(a + 2)a + (a + 2) · 2] + (−a)a + (−a) · 2 −(a2 + 2a + 2a + 4) − a2 − 2a −2a2 − 6a − 4 3. (10 points) Solve the following inequality for x. Then graph your solution on the real number line. 4x − 3 < 10 − 2(x − 1) Solution: 4x − 3 < 10 − 2(x − 1) 4x − 3 < −2x + 12 6x − 3 < 12 6x < 15 5 x< 2 4. (10 points) Plot (−1, 3) and (2, 4) on a rectangular coordinate system. Find the slope of the line passing through these two points. Solution: slope = y 2 − y1 4−3 1 = = x2 − x1 2 − (−1) 3 1 5. (10 points) Graph the solution of the following linear inequality: 3y ≥ 2x + 12 6. (10 points) Solve the following equation for x: | − x + 3| = 6 Solution: Separate into two cases: −x + 3 = 6 and −x + 3 = −6. Solve both equation and get x = −3 or x = 9. 7. (10 points) Find the equation of the line through (−1, 1) which is perpendicular to the line 3x + 4y = 5 Write your answer in slope-intercept form. Solution: Rewrite 3x + 4y = 5 in slope-intercept form: 3x + 4y = 5 4y = −3x + 5 3 5 y =− x+ 4 4 3 This tells us that the slope of the line 3x + 4y = 5 is − . So the slope of a 4 4 perpendicular line is . By point-slope form, the equation of the line asking in 3 the problem is 4 y − 1 = (x − (−1)) 3 Rewrite in slope-intercept form: y= 4 7 x+ 3 3 8. (10 points) One eighth of a number equals the number plus 14. What is the number? Solution: Assume the unknown number is x. Then 1 x = x + 14 8 Solve this equation x = 8x + 8 · 14 −7x = 8 · 14 x = −8 × 14 ÷ 7 x = −8 × 2 x = −16 In conclusion, the number satisfying the condition in the problem is −16. 9. (10 points) Solve the following system of equations for (x, y): ( 4x − y = 1 2x + y = 0 2 Solution: The sum of these two equations is 6x = 1 1 1 So we have x = . Substitute x into the second equation. We get 2 · + y = 0. 6 6 1 So y = − . In conclusion, the solution is 3 1 1 (x, y) = ( , − ) 6 3 10. (10 points) Describe the relation between the graph of f (x) = |x| and the graph of h(x) = |x−1|−1. Solution: Shifting the graph of f (x) = |x| one unit to the right gives the graph of g(x) = |x − 1|. Then shift the graph one unit downwards and get the graph of h(x) = |x − 1| − 1. 3