Add or subtract. Simplify your answer.
1. (x^2 + 8x)/ (x - 2)/ (3x + 14)/ (x - 2)
A: ?
2. (2t)/(4t^2) + (2)/(t)
A: ?
3. (m^2 - m - 2)/ (m^2 + 6m +5) - (2)/(m + 5)
A: ?
I suspect #1 is missing parentheses.
If it is a big fraction made up of two small fractions, it would read:
1.((x^2 + 8x)/(x - 2)) / ((3x + 14)/(x - 2))
Remember that parentheses are always there around the numerator and denominator.
These questions are solved using factors.
For number 1, factor as much as possible, and cancel common factors, but be sure to specify that the common factors cannot equal to zero. For example:
1. ((x^2 + 8x)/ (x - 2))÷((3x + 14)/ (x - 2))
= (x(x+8)/(x - 2))÷((3x + 14)/(x-2))
Now multiply the reciprocal of the denominator, instead of ÷
= (x(x+8)/(x-2)) * ((x-2)/(3x+14))
Then the common factor (x-2) can be cancelled if (x-2)≠0, or x≠2.
= x(x+8)/(3x+14) if x≠2
Attempt the other questions in a similar way.
To simplify the expressions, we'll follow the order of operations, which is usually abbreviated as PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction).
1. (x^2 + 8x)/ (x - 2)/ (3x + 14)/ (x - 2):
First, let's simplify each individual expression within the parentheses:
a) (x^2 + 8x)
b) (x - 2)
c) (3x + 14)
d) (x - 2)
To get the answer, we'll divide the expressions a, b, c, and d.
Step 1: Simplify expression a and b separately
- (x^2 + 8x) divided by (x - 2) simplifies to (x(x + 8))/(x - 2).
Step 2: Simplify expression c and d separately
- (3x + 14) divided by (x - 2) simplifies to (3(x + 14))/(x - 2).
Step 3: Divide both expressions obtained in step 1 and step 2:
- (x(x + 8))/(x - 2) divided by (3(x + 14))/(x - 2).
To divide fractions, we multiply by the reciprocal of the second fraction:
- (x(x + 8))/(x - 2) multiplied by (x - 2)/(3(x + 14))
- Simplifying this expression will give us the final answer.
2. (2t)/(4t^2) + (2)/(t):
This expression primarily involves addition and division. To simplify, let's convert each term into fractions with the same denominator.
Step 1: Simplify 2t/(4t^2)
- 2t divided by 4t^2 simplifies to (1/(2t)).
So we now have (1/(2t)) + (2)/(t).
Step 2: Getting the common denominator.
- The common denominator between 2t and t is 2t. So we rewrite the expression as follows:
- (1/(2t)) + (2)/(t) = (t + 4)/(2t).
This is the simplified answer.
3. (m^2 - m - 2)/ (m^2 + 6m + 5) - (2)/(m + 5):
This expression involves both addition and division. To simplify, we'll follow the same steps as before.
Step 1: Simplifying (m^2 - m - 2)/ (m^2 + 6m + 5)
- First, factorize both the numerator and denominator:
Numerator: (m - 2)(m + 1)
Denominator: (m + 5)(m + 1)
Thus, the expression becomes [(m - 2)(m + 1)] / [(m + 5)(m + 1)].
Step 2: Simplifying (2)/(m + 5)
- The expression remains unchanged.
Step 3: Getting the common denominator.
- Since the denominators are already the same, the expression becomes:
[(m - 2)(m + 1)] / [(m + 5)(m + 1)] - (2)/(m + 5)
Now, we have a common denominator and can combine the terms.
These three examples illustrate how to add or subtract and simplify expressions.