Questions: Solve the equation. (x+2)/3=1-(x+5)/7 The solution set is 3. (Type an integer or a fraction.)

Solve the equation. (x+2)/3=1-(x+5)/7 The solution set is 3. (Type an integer or a fraction.)
Transcript text: Solve the equation. \[ \frac{x+2}{3}=1-\frac{x+5}{7} \] The solution set is $\square$ 3. (Type an integer or a fraction.)
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Solution

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Solution Steps

Step 1: Eliminate Fractions

To eliminate the fractions, we first find a common denominator for the fractions on both sides of the equation. The denominators are 3 and 7, so the common denominator is 21. Multiply every term by 21 to clear the fractions:

\[ 21 \cdot \frac{x+2}{3} = 21 \cdot \left(1 - \frac{x+5}{7}\right) \]

This simplifies to:

\[ 7(x+2) = 21 - 3(x+5) \]

Step 2: Distribute and Simplify

Distribute the 7 on the left side and the 3 on the right side:

\[ 7x + 14 = 21 - 3x - 15 \]

Simplify the right side:

\[ 7x + 14 = 6 - 3x \]

Step 3: Combine Like Terms

Add \(3x\) to both sides to combine the \(x\) terms:

\[ 7x + 3x + 14 = 6 \]

This simplifies to:

\[ 10x + 14 = 6 \]

Subtract 14 from both sides to isolate the \(x\) term:

\[ 10x = 6 - 14 \]

\[ 10x = -8 \]

Step 4: Solve for \(x\)

Divide both sides by 10 to solve for \(x\):

\[ x = \frac{-8}{10} \]

Simplify the fraction:

\[ x = \frac{-4}{5} \]

Final Answer

The solution set is \(\boxed{\frac{-4}{5}}\).

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