range of (3x-2)/(x+5) — step-by-step solution

1) Let's find the real values taken by

\frac{3 x - 2}{x + 5}

, for real

x

⇥Let's find the domain of

\frac{3 x - 2}{x + 5}

⇥ For the expression

\frac{3 x - 2}{x + 5}

to be defined, we must have:

x + 5 \neq 0

⇥ⓘ We subtract

5

from both sides

⇥Solution found:

x \neq -5

The input domain is

x \neq -5

2) Set

y = \frac{3 x - 2}{x + 5}

and solve for

x

to express the input in terms of the attained value

⇥2.1) Let's solve in

ℝ

y = \frac{3 x - 2}{x + 5}

⇥

\frac{3 x - 2}{x + 5} = y

⇥We multiply both sides by

x + 5

, nonzero for the fractions to be defined

⇥

\frac{3 x - 2}{x + 5} \cdot (x + 5) = y \cdot (x + 5)

⇥ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{3 x - 2}{x + 5} \cdot (x + 5)

\frac{(3 x - 2) \cdot (x + 5)}{x + 5}

■The original denominator must remain nonzero. The fraction is simplified by dividing the numerator and the denominator by their common factor

x + 5

\frac{(3 x - 2) \cdot (x + 5)}{x + 5}

3 x - 2

⇥

3 x - 2 = y \cdot (x + 5)

⇥ⓘ Expand by multiplying each term inside the parentheses:

y \cdot (x + 5)

y x + y \cdot 5

⇥

3 x - 2 = y x + y \cdot 5

⇥ⓘ We add

2

to both sides

⇥

3 x = y x + y \cdot 5 + 2

⇥

3 x - y x = y \cdot 5 + 2

⇥

(3 - y) \cdot x = y \cdot 5 + 2

⇥ⓘ Both sides are divided by

(3 - y)

, which must be nonzero:

y \neq 3

⇥Candidates, pending the domain check:

x = \frac{5 y + 2}{3 - y}

⇥2.2) Let's evaluate the domain of

y = \frac{3 x - 2}{x + 5}

⇥ For the expression

\frac{3 x - 2}{x + 5}

to exist, you must check:

x + 5 \neq 0

⇥ⓘ We subtract

5

from both sides

⇥Domain constraints:

x \neq -5

3) The equation has a solution for

x

exactly when

y \neq 3

: these are the attained values

▶Range:

ℝ \ {3}