To solve the equation \(-5 + \frac{5}{2}x + 1 = x + 3\), we can start by simplifying it step by step.
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Combine the constants on the left side: \[ -5 + 1 = -4 \] So the equation becomes: \[ -4 + \frac{5}{2}x = x + 3 \]
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Now, let's isolate the term with \(x\). First, add 4 to both sides of the equation: \[ \frac{5}{2}x = x + 3 + 4 \] This gives us: \[ \frac{5}{2}x = x + 7 \]
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Next, subtract \(x\) from both sides: \[ \frac{5}{2}x - x = 7 \] To combine the terms on the left, recall that \(x\) can be written as \(\frac{2}{2}x\): \[ \frac{5}{2}x - \frac{2}{2}x = 7 \] This simplifies to: \[ \frac{3}{2}x = 7 \]
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To solve for \(x\), multiply both sides by the reciprocal of \(\frac{3}{2}\), which is \(\frac{2}{3}\): \[ x = 7 \times \frac{2}{3} = \frac{14}{3} \]
So, the value of \(x\) that makes the equation true is: \[ \boxed{\frac{14}{3}} \]