Verified Answer :

  • Multiply both sides by the reciprocal \(\frac{2}{3}\) to simplify the equation to \(2x + 5 = -5\).
  • Subtract 5 from both sides to isolate the term with \(x\), yielding \(2x = -10\).
  • Divide by 2 to find the solution.
  • The final solution is \(x = \boxed{-5}\).

Explanation

  1. Analysis of the Equation

The equation to solve is \(\frac{3}{2}(2x + 5) = -\frac{15}{2}\). This is a linear equation in one variable \(x\). To find the value of \(x\), we need to isolate it on one side of the equation. We observe that the term \((2x + 5)\) is multiplied by \(\frac{3}{2}\). Our strategy will be to use algebraic operations to isolate \(x\) step by step.

  1. Isolating the Expression with x

First, we can eliminate the coefficient \(\frac{3}{2}\) by multiplying both sides of the equation by its reciprocal, which is \(\frac{2}{3}\).
Multiplying the left side:
\(\frac{2}{3} \cdot \left(\frac{3}{2}(2x + 5)\right) = 2x + 5\)
Multiplying the right side:
\(\frac{2}{3} \cdot \left(-\frac{15}{2}\right) = -\frac{30}{6} = -5\)
Thus, the equation simplifies to:
\(2x + 5 = -5\)

  1. Solving for x

Now, we isolate the term \(2x\) by subtracting 5 from both sides of the equation:
\(2x = -5 - 5\)
\(2x = -10\)
Finally, we solve for \(x\) by dividing both sides by 2:
\(x = \frac{-10}{2} = -5\)

Examples

Equations like this represent fundamental relationships between variables. Imagine a scenario where you are tracking the temperature change in a controlled experiment. If the temperature is modeled by a linear expression such as \(\frac{3}{2}(2x + 5)\), where \(x\) represents time in minutes, and you need to find the specific time \(x\) when the temperature reaches a target value of \(-\frac{15}{2}\) degrees, solving this linear equation provides the exact time required. This type of algebraic balancing is crucial in engineering, finance, and daily planning tasks where one must adjust variables to reach a target outcome.

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