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Chapter 6: Problem 92

Solve each equation. Use set notation to express solution sets for equations with no solution or equations that are true for all real numbers. \(\frac{x}{2}+\frac{2 x}{3}+3=x+3\)

Short Answer

Expert verified
The solution to the equation is \(x = 0\).

Step by step solution

01

Simplify Both Sides of The Equation

To simplify both sides, we will start by distributing the denominator in the left side of the equation: \(\frac{x}{2}+\frac{2 x}{3}= \frac{3x}{6} + \frac{4x}{6}\) which simplifies to \(\frac{7x}{6}\). Thus the equation can be rewritten as \( \frac{7x}{6} + 3= x+3 \)
02

Cancel Out The Same Terms on Both Sides

We see that there is '+3' on both sides of the equation. We can cancel out these terms by subtracting 3 from both sides. Doing this, the equation simplifies to \(\frac{7x}{6}= x\)
03

Subtract The Fractions

Next we subtract \( \frac{7x}{6}\) from both sides to isolate our variable, x. The equation then becomes \(0 = x-\frac{7x}{6}\) or \(0= \frac{-x}{6}\)
04

Solve The Resulting Equation

Solving the resulting equation gives us \(x = 0\). Therefore, our solution is \(x = 0\).

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Most popular questions from this chapter

Solve each equation by the method of your choice. \(2 x^{2}-9 x-3=9-9 x\)

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