Question

solve the equation. check each solution to verify the solution is valid.

1. $\frac{2x}{5} + \frac{1}{2} = \frac{3x}{10}$

answer: $\boldsymbol{x=-5}$ $x\
eq10$
$4x + 5 = 3x \
ightarrow 4x-3x+5=0$
$x + 5 = 0$
$x=-5$

1. $\frac{3}{x} + \frac{1}{2x} = \frac{7}{8}$

answer: _______________

Explanation:

Step1: Identify LCD of denominators

The least common denominator (LCD) of $x$, $2x$, and $8$ is $8x$.

Step2: Multiply all terms by LCD

$$8x \cdot \frac{3}{x} + 8x \cdot \frac{1}{2x} = 8x \cdot \frac{7}{8}$$
Simplify each term: $24 + 4 = 7x$

Step3: Combine constant terms

$$28 = 7x$$

Step4: Solve for $x$

$$x = \frac{28}{7} = 4$$

Step5: Verify validity

Substitute $x=4$ into original equation: $\frac{3}{4} + \frac{1}{8} = \frac{6}{8} + \frac{1}{8} = \frac{7}{8}$, which matches the right-hand side. Also, $x=4$ does not make any original denominator equal to 0, so it is valid.

Answer:

$x=4$, $x
eq0$