Section 10.2 Homework

Question 7, R.4.27

Solve the equation:

[tex]\[ \frac{x+8}{3} = \frac{x-3}{2} \][/tex]

Select the correct choice below and, if necessary, fill in the answer box:

A. The solution(s) is/are [tex]\( x = \square \)[/tex]
(Simplify your answer. Use a comma to separate answers as needed.)

B. There is no solution.



Answer :

To solve the equation [tex]\(\frac{x + 8}{3} = \frac{x - 3}{2}\)[/tex], follow these steps:

1. Clear the Fractions:
To eliminate the fractions, multiply both sides of the equation by the least common multiple (LCM) of the denominators, which is 6.

[tex]\[ 6 \cdot \frac{x + 8}{3} = 6 \cdot \frac{x - 3}{2} \][/tex]

2. Simplify the Equation:
Multiply through and simplify:

[tex]\[ 6 \cdot \frac{x + 8}{3} = (6 \div 3) \cdot (x + 8) = 2(x + 8) \][/tex]
[tex]\[ 6 \cdot \frac{x - 3}{2} = (6 \div 2) \cdot (x - 3) = 3(x - 3) \][/tex]

So, the equation simplifies to:

[tex]\[ 2(x + 8) = 3(x - 3) \][/tex]

3. Distribute:
Distribute the constants through the parentheses:

[tex]\[ 2x + 16 = 3x - 9 \][/tex]

4. Combine Like Terms:
First, get all x terms on one side by subtracting 2x from both sides:

[tex]\[ 16 = x - 9 \][/tex]

Then add 9 to both sides to isolate x:

[tex]\[ 16 + 9 = x \][/tex]

[tex]\[ 25 = x \][/tex]

5. State the Solution:
The solution to the equation is:

[tex]\[ x = 25 \][/tex]

Therefore, the correct choice is:

A. The solution is [tex]\( x = 25 \)[/tex].