Answer
The solution is:
$$x = 2$$
To check if the solution is correct, we plug $2$ in for $x$ into the original equation:
$$\frac{2}{2} - \frac{1}{10} = \frac{2}{5} + \frac{1}{2}$$
Simplify:
$$1 - \frac{1}{10} = \frac{2}{5} + \frac{1}{2}$$
Multiply by the least common denominator, which is $10$ in this case, to get rid of fractions:
$$10(1) - 10(\frac{1}{10}) = 10(\frac{2}{5}) + 10(\frac{1}{2})$$
Divide out common factors to get rid of fractions:
$$10 - 1 = 4 + 5$$
Simplify:
$$9 = 9$$
Both sides of the equation are equal, so the solution is correct.
Work Step by Step
We want to multiply all terms by their least common denominator, which is $10$ in this case, so we don't have to deal with fractions.
$$10(\frac{x}{2}) - 10(\frac{1}{10}) = 10(\frac{x}{5}) + 10(\frac{1}{2})$$
We divide out common factors to get rid of the fractions:
$$5x - 1 = 2x + 5$$
We subtract $2x$ from both sides to isolate the variable:
$$3x - 1 = 5$$
We add $1$ to both sides to isolate the variable:
$$3x = 6$$
Divide by $3$ on both sides to isolate $x$:
$$x = 2$$
To check if the solution is correct, we plug $2$ in for $x$ into the original equation:
$$\frac{2}{2} - \frac{1}{10} = \frac{2}{5} + \frac{1}{2}$$
Simplify:
$$1 - \frac{1}{10} = \frac{2}{5} + \frac{1}{2}$$
Multiply by the least common denominator, which is $10$ in this case, to get rid of fractions:
$$10(1) - 10(\frac{1}{10}) = 10(\frac{2}{5}) + 10(\frac{1}{2})$$
Divide out common factors to get rid of fractions:
$$10 - 1 = 4 + 5$$
Simplify:
$$9 = 9$$
Both sides of the equation are equal, so the solution is correct.