x/5 =2+(x+2)/2 — step-by-step solution

1) Let's solve in

ℝ

\frac{x}{5} = 2 + \frac{x + 2}{2}

ⅰ

2 + \frac{x + 2}{2}

\frac{2 \cdot 2 + (x + 2)}{2}

\frac{x}{5} = \frac{2 \cdot 2 + (x + 2)}{2}

ⓘ Multiplications are done before additions and subtractions:

2 \cdot 2

4

\frac{x}{5} = \frac{4 + x + 2}{2}

ⓘ The numerator and the denominator are computed before the division:

4 + 2

6

\frac{x}{5} = \frac{6 + x}{2}

2) Let's sum the fractions:

\frac{x}{5} - \frac{6 + x}{2} = 0

ⓘ To add fractions, put them over the same denominator, then add the numerators:

\frac{x}{5} - \frac{6 + x}{2}

\frac{x \cdot 2 - (6 + x) \cdot 5}{5 \cdot 2}

\frac{x \cdot 2 - (6 + x) \cdot 5}{5 \cdot 2} = 0

ⅰ

5 \cdot 2

10

\frac{x \cdot 2 - (6 + x) \cdot 5}{10} = 0

Let's solve in

ℝ

x \cdot 2 - (6 + x) \cdot 5 = 0

ⓘ Expand by multiplying each term inside the parentheses:

- (6 + x) \cdot 5

- (30 + x \cdot 5)

x \cdot 2 - (30 + x \cdot 5) = 0

ⓘ A minus sign before a parenthesis changes the sign of each of its terms:

- (30 + x \cdot 5)

- 30 - x \cdot 5

x \cdot 2 - 30 - x \cdot 5 = 0

ⓘ Like terms are grouped by adding their coefficients:

x \cdot 2 - x \cdot 5

-3 x

- 3 x - 30 = 0

ⓘ We add

30

to both sides

- 3 x = 30

ⓘ Both sides are divided by

-3

x = \frac{30}{-3}

ⓘ The sign of a fraction goes in front of the bar, whether it comes from the numerator or the denominator:

\frac{30}{-3}

- \frac{30}{3}

x = - \frac{30}{3}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

3

, their greatest common divisor:

- \frac{30}{3}

-10

▶Solution found:

x = -10