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Chapter 6: Problem 53

Solve and check each equation. \(\frac{3 x}{4}-3=\frac{x}{2}+2\)

Short Answer

Expert verified
The solution to the equation is \(x = 20\).

Step by step solution

01

Clear the fractions

To avoid working with fractions, it can be easier to clear the fractions by multiplying every term in the equation by the denominator. Here, the LCD of 2 and 4 is 4. Therefore, multiply every term by 4 to get rid of fractions: \(4 \cdot \frac{3 x}{4}-4 \cdot 3=4 \cdot \frac{x}{2}+4 \cdot 2\). This simplifies to \(3x - 12 = 2x + 8\).
02

Combine like terms

Combine like terms on each side by subtracting 2x from both sides to keep x terms on one side: \(3x - 2x - 12 = 2x - 2x + 8\). Simplifying we get \(x - 12 = 8\).
03

Solve for x

Solve for x by adding 12 to both sides to isolate the 'x': \(x = 8 + 12\). Thus, we get \(x = 20\).
04

Check the solution

Substitute \(x = 20\) into the original equation to check if rhs = lhs. \(\frac{3(20)}{4}-3 = \frac{20}{2}+2 \), simplifying we get \(15 - 3 = 10 + 2 \). So \(12 = 12\), which is true. Therefore, the calculated 'x' value is correct.

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