To solve the equation \( 5(2x + 2) - 4 = 2x + 6 \), we start by simplifying both sides step-by-step.
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Distribute the 5 on the left side: \[ 5(2x + 2) = 10x + 10 \] So, the left side becomes: \[ 10x + 10 - 4 = 10x + 6 \]
Now, the equation looks like this: \[ 10x + 6 = 2x + 6 \]
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Next, we want to isolate the \( x \) terms. Subtract \( 2x \) from both sides: \[ 10x - 2x + 6 = 6 \] which simplifies to: \[ 8x + 6 = 6 \]
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Now, subtract 6 from both sides: \[ 8x + 6 - 6 = 6 - 6 \] This results in: \[ 8x = 0 \]
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Finally, divide both sides by 8: \[ x = 0 \]
Now, we can check the original equation to determine if it has a unique solution, infinite solutions, or no solutions:
- Substitute \( x = 0 \) back into the original equation: \[ 5(2(0) + 2) - 4 = 2(0) + 6 \] Simplifying both sides yields: \[ 5(2) - 4 = 6 \implies 10 - 4 = 6 \implies 6 = 6 \]
Since both sides of the equation are equal when \( x = 0 \), we have verified that there is a single solution.
The conclusion is: The equation has one solution: \( x = 0 \).