Download solutions for the midterm

Math 084.03 - Midterm Solutions
(1) For each equation below, solve for x.
(a) 2(x + 1) + 1 = 2x + 3
2(x + 1) + 1 = 2x + 3
2x + 2 + 1 = 2x + 3
2x + 3 = 2x + 3
0 = 0.
All values of x work in this equation, so every number is a solution.
(b) 4(x − 1) = 2x + (2x + 2)
4(x − 1) = 2x + (2x + 2)
4x − 4 = 4x + 2
−4 = 2.
There is no value of x that makes this equation true, so this equation has no solutions.
(c) 2x − 3y = 5x + 6
2x − 3y = 5x + 6
2x − 3y − 5x = 6
−3x − 3y = 6
−3x = 3y + 6
3y + 6
x=
−3
x = −y − 2.
2
(2) Simplify the expression below. For each step, state what property of numbers justifies that
step.
−2x(x + 1) − 4(−3x − 2) + 3x − 4
−2x(x + 1) − 4(−3x − 2) + 3x − 4
−2x2 − 2x + 12x + 8 + 3x − 4
distribution
−2x2 − 2x + 12x + 3x + 8 − 4
commutativity of addition
2
−2x + 13 + 4
factoring and arithmetic
Math 084.03
(3) Solve the inequality below, and draw a picture of the collection of solutions.
2(z − 3) − 7 ≥ 5z + 12
2(z − 3) − 7 ≥ 5z + 12
2z − 6 − 7 ≥ 5z + 12
2z − 13 ≥ 5z + 12
2z ≥ 5z + 25
−3z ≥ 25
z ≤ − 25
3 .
The solution set for this inequality is below:
3
4
(4) Two plumbers are bidding on a job. The first estimates that the job will cost $600 in
materials plus $30 per hour of labor. The second plumber estimates that the job will cost
$700 in materials plus $28 per hour of labor. How many hours of labor would make the
two plumbers charge the same for the job?
If the plumbers work for h hours, the first plumber charges
$30
$600 +
h,
1hour
and the second plumber charges
$28
$700 +
h.
1hour
We want these two to be the same, so we declare them to be equal:
600 + 30h = 700 + 28h.
Now we can solve this equation for h.
600 + 30h = 700 + 28h
600 + 2h = 700
2h = 100
h = 50.
So the two plumbers charge the same amount for 50 hours.
Math 084.03
5
(5) Consider the equation
4x + 3y = 6.
(a) Find the x intercept, y intercept, and one other point that is on this line. State your
results as points (that is, as pairs of numbers).
The x intercept is where y is 0, so put y = 0 into the equation and find x:
4x + 3 · 0 = 6
⇒
x = 32 .
So the x intercept is the point ( 32 , 0). The y intercept is where x is 0, so put x = 0
into the equation:
4 · 0 + 3y = 6
⇒
y = 2.
So the y intercept is (0, 2). To find another point, we can use any number for x, then
use the equation to find the corresponding y value. I’ll use x = 1.
4 · 1 + 3y = 6
⇒
3y = 2
So the point (1, 23 ) is also on the line.
(b) Graph the line.
⇒
y = 23 .
6
(6) (a) Is the point (400, 321) on the line 32x − 49y = 50? Briefly explain why your answer is
correct.
If a point is on a line, then the point satisfies the equation of the line. We can check
if the point (400, 321) satisfies the equation 32x − 49y = 50 just by putting in the
coordinates of the point for x and y:
32 · 400 − 49 · 321 = −2929 6= 50.
Since the point does not satisfy the equation, the point is not on the line.
(b) Circle any equations below that are linear:
2(x − 1) = 6(y + x)
linear!
7x + x(y − 3) = 2x + y
46
x+
y = − 155
10012
6341
linear!