Download solutions for the old midterm

(1) Using variables a, b, and c, write equations that express the following properties of real
numbers.
(a) The associativity of addition.
(a + b) + c = a + (b + c)
(b) The commutativity of multiplication.
a·b=b·a
(c) The distribution of multiplication over addition.
a · (b + c) = a · b + a · c
(d) The commutativity of addition.
a+b=b+a
(e) The associativity of multiplication.
(a · b) · c = a · (b · c)
2
(2) (a) Simplify the equation
−2x(3x − 4) = 5(2y + 5).
−2x(3x − 4) = 5(2y + 5)
(−2x) · (3x) − (−2x) · 4 = 10y + 25
−6x2 + 8x = 10y + 25
If we want, we could also solve for y:
−6x2 + 8x = 10y + 25
−6x2 + 8x − 25 = 10y
6 2
− 10
x +
8
10
−
25
10
= y.
(b) Is the equation from (a) linear? Briefly explain why or why not.
No, this equation is not linear. An equation is linear if we can put in the form
Ax + By = C, where A, B, and C are constants. But the equation from part (a)
has an x2 term, which linear equations do not.
Math 084.03
(3) Solve the following inequality, and draw a diagram of the set of solutions:
−2(a + 3) − 3(−a + 12) ≥ 10a + 20.
−2(a + 3) − 3(−a + 12) ≥ 10a + 20
−2a − 6 + 3a − 36 ≥ 10a + 20
−9a − 42 ≥ 20
−9a ≥ 62
a ≤ − 62
9 .
This is the collections of a values pictured below.
3
4
(4) For each of the following, decide if the chain of reasoning is correct or not. Circle any
errors.
(a) Simplifying:
(x − 5) · (x + 2)
(x − 5) · x + (x − 5) · 2
x·x+5·x+2·x−5·2
the “+5” on this line should be “−5”
x2 + (5 + 2) · x − 10
x2 + 7x − 10
This solution is incorrect because of the mistake on the third line.
(b) Solving an inequality:
2a + 3(−a + 3) ≤ 12
2a + (−3)a + 9 ≤ 12
(2 − 3)a + 9 ≤ 12
−a + 9 ≤ 12
−a + 9 + a ≤ 12 + a
9 ≤ 12 + a
9 − 12 ≤ 12 + a − 12
−3 ≤ a
This solution is correct. This is not the only way to do this problem, but it is correct!
Math 084.03
5
(5) Solve the equations for x:
(a) 4x − 13x + 1 = −7
4x − 13x + 1 = −7
−9x + 1 = −7
−9x = −8
x=
−8
−9
x=
8
9
So the only solution to this equation is when x = 8/9.
(b) 2x + 3 = 2(x + 1)
2x + 3 = 2(x + 1)
3=2
1 = 0.
There are no x-values that make this equation true. I.e. there are no solutions.