Solved on Jan 21, 2024

Find the solution set to the equation 3a+5=3a7|3a+5| = |3a-7|.

STEP 1

Assumptions
1. We are given an absolute value equation 3a+5=3a7|3a + 5| = |3a - 7|.
2. We need to find the solution set for the variable aa.
3. The absolute value of a number is always non-negative.
4. The equation x=y|x| = |y| holds if and only if x=yx = y or x=yx = -y.

STEP 2

We will consider the two possible cases that arise from the property of absolute values mentioned in STEP_1.4.
Case 1: The expressions inside the absolute values are equal. 3a+5=3a73a + 5 = 3a - 7
Case 2: The expressions inside the absolute values are negatives of each other. 3a+5=(3a7)3a + 5 = -(3a - 7)

STEP 3

Solve for aa in Case 1.
3a+5=3a73a + 5 = 3a - 7

STEP 4

Subtract 3a3a from both sides of the equation to isolate the constant terms.
3a+53a=3a73a3a + 5 - 3a = 3a - 7 - 3a

STEP 5

Simplify both sides of the equation.
5=75 = -7

STEP 6

We observe that the equation 5=75 = -7 is not true for any value of aa. Therefore, there are no solutions from Case 1.

STEP 7

Now, solve for aa in Case 2.
3a+5=(3a7)3a + 5 = -(3a - 7)

STEP 8

Distribute the negative sign on the right side of the equation.
3a+5=3a+73a + 5 = -3a + 7

STEP 9

Add 3a3a to both sides of the equation to get all terms with aa on one side.
3a+5+3a=3a+7+3a3a + 5 + 3a = -3a + 7 + 3a

STEP 10

Simplify both sides of the equation.
6a+5=76a + 5 = 7

STEP 11

Subtract 5 from both sides of the equation to isolate the terms with aa.
6a+55=756a + 5 - 5 = 7 - 5

STEP 12

Simplify both sides of the equation.
6a=26a = 2

STEP 13

Divide both sides of the equation by 6 to solve for aa.
6a6=26\frac{6a}{6} = \frac{2}{6}

STEP 14

Simplify both sides of the equation.
a=13a = \frac{1}{3}

STEP 15

Since we found a solution in Case 2 and no solution in Case 1, the solution set for the equation 3a+5=3a7|3a + 5| = |3a - 7| is {13}\left\{\frac{1}{3}\right\}.
The correct choice is A. The solution set is {13}\left\{\frac{1}{3}\right\}.

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