Add: 1/2(x+4)+1/4(4x−2). Answer should be in mixed number form.(1 point)
1/2(x+4) + 1/4(4x-2) can be simplified as follows:
= (1/2)x + 2 + (1/4)(4x) - (1/4)(2)
= (1/2)x + 2 + x - 1/2
= (1/2)x + x + 2 - 1/2
= (3/2)x + 3/2
So, the answer in mixed number form is 1 (1/2)x + 1 (1/2).
To add the given expressions, first simplify them individually and then add them together.
First, let's simplify 1/2(x + 4):
1/2(x + 4) = 1/2 * x + 1/2 * 4 = 1/2x + 2
Next, let's simplify 1/4(4x - 2):
1/4(4x - 2) = 1/4 * 4x + 1/4 * (-2) = x - 1/2
Now, let's add the two simplified expressions:
(1/2x + 2) + (x - 1/2) = 1/2x + x + 2 - 1/2
To combine like terms, let's find a common denominator for 1/2x and x:
The common denominator of 1/2 and 1 is 2, so multiply 1/2x by 2/2 to get 2/4x.
Now, the expression becomes:
2/4x + x + 2 - 1/2
Combining like terms:
2/4x + x + 2 - 1/2 = 2/4x + x + 4/2 - 1/2 = (2 + 4)/2 + x + 2/4x - 1/2 = 6/2 + x + 2/4x - 1/2 = 3 + x + 1/2x - 1/2
To add the fraction terms, let's find a common denominator for 1/2x and -1/2:
The common denominator of 2x and 2 is 2x, so we multiply 1/2x by x/x to get x/2x.
Now, the expression becomes:
3 + x + 1/2x - 1/2 = 3 + x + x/2x - 1/2 = 3 + x + x/2x - 1/2 = 3 + x + x/2x - (1*2)/(2*2x) = 3 + x + x/2x - 2/4x = 3 + x + x/2x - 1/2x = 3 + x(1 + 1/2 - 1/2) = 3 + x(1) = 3 + x
Therefore, the final answer, in mixed number form, is: 3 + x.
To solve this problem, we need to simplify the expression by performing the operations within the parentheses and then adding the terms.
Step 1: Simplify within the parentheses
- For the first term, 1/2(x + 4), we distribute 1/2 to each term inside the parentheses:
1/2(x) + 1/2(4) = 1/2x + 2/2 = 1/2x + 1.
- For the second term, 1/4(4x - 2), we distribute 1/4 to each term inside the parentheses:
1/4(4x) + 1/4(-2) = 1x + (-1/2) = x - 1/2.
Step 2: Combine like terms
Now that we have simplified the expression, we can add the terms together:
(1/2x + 1) + (x - 1/2) = 1/2x + x + 1 - 1/2.
Step 3: Find the common denominator
In order to add fractions together, we need to have a common denominator. In this case, 1/2x + x already have a common denominator of 2, but we need to adjust 1 to have the same denominator:
1 = 2/2.
Step 4: Add the fractions together
Now, we can add the fractions:
(1/2x + x + 1 - 1/2) = 1/2x + 2/2x + 2 - 1/2 = (1/2 + 2/2)x + (2 - 1/2) = (3/2)x + (4/2 - 1/2).
Step 5: Simplify further
(3/2)x + (4/2 - 1/2) = (3/2)x + (3/2) = (3/2)x + 3/2 = (x + 3)/2.
The simplified expression, in mixed number form, is (x + 3)/2.