These are six questions and six answers
Question 1. Translate y = 2/x 3 units to the left and 4 units up.
Answer:
2
y = -------- + 4 <------- the fourth option
(x + 3)
Explanation:
1) Given function:
y = f(x) = 2 / x
2) Translatiing 3 units to the left is making f(x + 3), so that implies:
y = 2 / (x + 3)
3) Translating 4 units up is making f(x + 3) + 4, so that implies:
2
y = -------- + 4
(x + 3)
Which is the fourth option.
Question 2. simplify
t^2 + 2t - 24
----------------
t^2 - 36
Answer: fourth option
t - 4
------ , with t ≠ - 6 and t ≠ 6.
t + 6
Explanation:
1) Factor the numerator:
t^2 + 2t - 24 = (t + 6) (t - 4)
2) Factor the denominator:
t^2 - 36 = (t + 6) (t - 6)
3) Rewrite the fraction:
(t + 6) (t - 4)
-------------------
(t + 6) (t - 6)
4) Cancel the factor t + 6 which is in both numerator and denominator, which you can do only y t + 6 ≠ 0 => t ≠ -6.
t - 4
-------
t - 6
That is the simplified expression, with the restrictions that t ≠ - 6 and t ≠ 6, because the denominator cannot be 0.
Question 3. Find the product of:
x^2 + 7x + 10 x^2 - 3x - 18
------------------ * ---------------------
x + 3 x^2 + x - 2
Answer: the fhird option:
(x + 5) (x - 6)
------------------
x - 1
with x ≠ -3, x ≠ - 2, and x ≠ 1.
Explanation:
1) Factor x^2 + 7x + 10
x^2 + 7x + 10 = (x + 5) (x + 2)
2) Factor x^2 - 3x - 18
x^2 - 3x - 18 = (x - 6)(x + 3)
3) Factor x^2 + x - 2
x^2 + x - 2 = (x + 2) (x - 1)
4) Rewrite the given expression using the factors:
(x + 5) (x + 2) (x - 6) (x + 3)
--------------------------------------
(x + 3) (x + 2) (x - 1)
5) Cancel the factors that appear on both the numerator and denominator:
(x + 5) (x - 6)
------------------
x - 1
The restrictions are those values of x that make any factor that is or was in the denominator: x ≠ -3, x ≠ - 2, and x ≠ 1.
Question 4. Simplify the complex fraction:
n - 4
------------------
n^2 - 2n - 15
--------------------------
n + 1
-----------
n + 3
Answer: option 4.
n - 4
------------------
(n - 5) ( n + 3)
Explanation
1) Factor n^2 - 2n - 15
n^2 - 2n - 15 = (n - 5)(n + 3)
2) rewrite the expression:
n - 4
---------------------
(n - 5) ( n + 3)
----------------------------
n + 1
----------
n + 3
3) Convert (n + 1) / (n + 3) into its reciprocal (n + 3) / (n + 1), and multiply instead of dividing.
n - 4 n + 3
--------------------- * --------
(n - 5) ( n + 3) ( n + 1)
4) Cancel n + 3
n - 4
------------------
(n - 5) ( n + 3)
That is the simplest form.
Question 5. Find the difference
Answer: second choice
1 - n
--------
n + 4
Explanation:
1) given:
n^2 + 3n + 2 2n
----------------- - -------
n^2 + 6n + 8 n + 4
2) factor the two quadratic trinomials
n^2 + 3n + 2 = (n + 2) ( n + 1)
n^2 + 6n + 8 = (n + 4) (n + 2)
3) Rewrite the expression:
(n + 2) (n + 1) 2n
----------------------- - -------
(n + 4 ) (n + 2) n + 4
4) cancel the factor n + 2
n + 1 2n
----------- - --------
n + 4 n + 4
5) subract the fractions. They have the same denominator.
n + 1 - 2n 1 - n
--------------- = ----------
n + 4 n + 4
And that is the simples form.
Question 6. Problem
Irina paints 1 room in 9 hours
Paulo paints 1 room in 8 hours
How long working together?
Answer:
x / 9 + x / 8 = 1, x ≈ 4.24 hours
Explanation:
1) In the time x, Irina will paint x / 9 parts of the room
2) Paulo will paint (in the same time) x / 8 parts of the same room
3) The total painted is 1 room.
So the equation is:
x / 9 + x / 8 = 1
4) The solution of that equation is:
[8x + 9x ] / (9*8) = 1
=> 8x + 9x = 72
=> 17x = 72
=> x = 72 / 17
=> x ≈ 4.24 hours
