range of (-5)/(2x+7) — step-by-step solution

1) Let's find the real values taken by

- \frac{5}{2 x + 7}

, for real

x

⇥Let's find the domain of

- \frac{5}{2 x + 7}

⇥ For the expression

- \frac{5}{2 x + 7}

to exist, you must check:

2 x + 7 \neq 0

⇥ⓘ We subtract

7

from both sides

⇥

2 x \neq -7

⇥ⓘ Both sides are divided by

2

⇥Solution obtained:

x \neq - \frac{7}{2}

The input domain is

x \neq - \frac{7}{2}

2) Set

y = - \frac{5}{2 x + 7}

and solve for

x

to express the input in terms of the attained value

⇥2.1) Let's solve in

ℝ

y = - \frac{5}{2 x + 7}

⇥

- \frac{5}{2 x + 7} = y

⇥We multiply both sides by

2 x + 7

, nonzero for the fractions to be defined

⇥

- \frac{5}{2 x + 7} \cdot (2 x + 7) = y \cdot (2 x + 7)

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

- \frac{5}{2 x + 7} \cdot (2 x + 7)

- \frac{5 \cdot (2 x + 7)}{2 x + 7}

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

- \frac{5 \cdot (2 x + 7)}{2 x + 7}

-5

⇥

-5 = y \cdot (2 x + 7)

⇥ⓘ Expand by multiplying each term inside the parentheses:

y \cdot (2 x + 7)

y \cdot 2 x + y \cdot 7

⇥

-5 = y \cdot 2 x + y \cdot 7

⇥

y \cdot 2 x + y \cdot 7 = -5

⇥

y \cdot 2 x = - 5 - y \cdot 7

⇥ⓘ Both sides are divided by

y \cdot 2

, which must be nonzero:

y \neq 0

⇥

x = \frac{- 5 - y \cdot 7}{y \cdot 2}

⇥ⓘ The minus sign is factored out, changing the sign of each term:

- 5 - y \cdot 7

- (5 + y \cdot 7)

⇥

x = - \frac{5 + y \cdot 7}{y \cdot 2}

⇥Candidates, pending the domain check:

x = - \frac{7 y + 5}{2 y}

⇥2.2) Let's evaluate the domain of

y = - \frac{5}{2 x + 7}

⇥ For the expression

- \frac{5}{2 x + 7}

to exist, you must check:

2 x + 7 \neq 0

⇥ⓘ We subtract

7

from both sides

⇥

2 x \neq -7

⇥ⓘ Both sides are divided by

2

⇥Domain constraints:

x \neq - \frac{7}{2}

3) The equation has a solution for

x

exactly when

y \neq 0

: these are the attained values

▶Range:

ℝ \ {0}