domain of f(x)=5/(2sqrt(5x+6)) — step-by-step solution

We save the definition:

f (x) = \frac{5}{2 \cdot \sqrt{5 x + 6}}

1) Let's evaluate the domain of

\frac{5}{2 \cdot \sqrt{5 x + 6}}

For the expression

\frac{5}{2 \cdot \sqrt{5 x + 6}}

to be defined, we must have:

2 \cdot \sqrt{5 x + 6} \neq 0

For the expression

\sqrt{5 x + 6}

to exist, you must check:

5 x + 6 \geq 0

2) Let's solve in

ℝ

2 \cdot \sqrt{5 x + 6} \neq 0

We simplify the expression by

2

⇥Let's solve in

ℝ

\sqrt{5 x + 6} \neq 0

⇥

\sqrt{5 x + 6} \neq 0

is equivalent to:

5 x + 6 > 0

⇥ⓘ We subtract

6

from both sides

⇥

5 x > -6

⇥ⓘ Both sides are divided by

5

⇥Solution found:

x > - \frac{6}{5}

3) Let's solve in

ℝ

5 x + 6 \geq 0

ⓘ We subtract

6

from both sides

5 x \geq -6

ⓘ Both sides are divided by

5

Solution obtained:

x \geq - \frac{6}{5}

Intersecting

x > - \frac{6}{5}

and

x \geq - \frac{6}{5}

gives

x > - \frac{6}{5}

▶Domain of definition:

x > - \frac{6}{5}