Download Math 1031 Sample Midterm 1 Solutions 1. Factor completely. x3

Math 1031
Sample Midterm 1 Solutions
1. Factor completely.
x3 − x2 y − 9x + 9y
(a) (x + 3)(x − 3)(x + y)
(b) (x2 + 9)(x − y)
(c) (x + 9y)(x2 − xy + 9)
(d) (x + 3)(x − 3)(x − y)
Solution: d
2. Find the exact solution to the numerical expression.
(4−2 + 3−1 )−1
(a)
19
48
(b) 25
1
25
(d) 48
19
(c)
Solution: d
3. Solve the inequality. Express the solution in interval notation.
x
20 + x
<
3
15
(a) (−∞, 5)
(b) (5, ∞)
(c) (−∞, −2)
(d) (−2, ∞)
Solution: a
1
4. Solve the equation.
√
3
7x + 1 = −4
, − 65
}
(a) x = { 65
7
7
(b) x =
(c) x =
65
7
− 65
7
(d) no real solutions
Solution: c
5. Solve for a.
2(a + 3) = −3(2a − 5)
(a) a =
(b) a =
−21
8
9
8
(c) a = −1
(d) a =
−9
4
Solution: b
6. Subtract and simplify if possible.
n2
(a)
−13−5n
(n−3)(n+3)
(b)
−3
(n−3)(n+3)
5
2
−
−9 n−3
−3
n2 −n−12
17−5n
(d) (n−3)(n+3)
(c)
Solution: a
7. A total of $4, 000 is invested, part at 4% interest per year, and the
rest at 5% interest per year. If the total yearly interest is $187.50, how
much is invested at each rate?
Solution: $1, 250 invested at 4% interest and $2, 750 invested at 5%
interest
8. Find the real solutions for t in the following equation.
t−4 − 2t−2 − 15 = 0
√
Solution: t = {−
√
5
5
,
}
5
5
2
9. A car radiator has 5 quarts of 10% antifreeze. How much should
be drained and replaced with 100% antifreeze to get 5 quarts of 40%
antifreeze?
Solution: 5/3 quarts
10. Solve the inequality and express the solution set in interval notation.
4x − 1
≤6
x−1
Solution: [5/2, ∞)
11. A rectangle is 5 times as long as it is wide. Its area is 125 square feet.
(a) Write an equation to describe this situation.
Solution: x · 5x = 125
(b) Solve the equation.
Solution: x = {−5, 5}
(c) What are the dimensions of the rectangle?
Solution: The rectangle is 5 feet by 25 feet.
12. Find the values of x that satisfy the inequality. Express the solution
in interval notation.
3x − 1 >5
3
2 Solution: (−∞, −7/9) ∪ (13/9, ∞)
3