-2*sqrt(x-1)+4>0 — step-by-step solution

Let's solve in

ℝ

- 2 \cdot \sqrt{x - 1} + 4 > 0

Since

-2 < 0

, divide both sides by

-2

and reverse the inequality:

\sqrt{x - 1} - 2 < 0

Isolate the square root:

\sqrt{x - 1} < 2

. Since both sides are nonnegative, square both sides, which keeps the direction of the inequality:

x - 1 < 2^{2}

ⓘ An integer power is a repeated product:

2^{2}

2 \cdot 2

4

x - 1 < 4

ⓘ We add

1

to both sides

x < 4 + 1

ⓘ Start from

4

and move forward

1

on the number line:

4 + 1

5

Candidates, pending the domain check:

x < 5

Let's find the domain of

- 2 \cdot \sqrt{x - 1} + 4 > 0

For the expression

\sqrt{x - 1}

to be defined, we must have:

x - 1 \geq 0

ⓘ We add

1

to both sides

Domain constraints:

x \geq 1

▶Solution:

x \in [1, 5)