Question

graph the solution to the inequality. (-5sqrt{x} - 2 < -10)

Explanation:

Step1: Isolate the square root term

First, we start with the inequality \(-5\sqrt{x}-2 < -10\). Add \(2\) to both sides of the inequality:
\[

$$\begin{align*} -5\sqrt{x}-2 + 2&< -10 + 2\\ -5\sqrt{x}&< -8 \end{align*}$$

\]

Step2: Divide by -5 (and reverse inequality)

Divide both sides by \(-5\). Remember that when we divide or multiply an inequality by a negative number, the direction of the inequality sign reverses. So we get:
\[
\sqrt{x}>\frac{-8}{-5}=\frac{8}{5} = 1.6
\]
Wait, no, wait. Wait, the original inequality is \(-5\sqrt{x}-2 < -10\). Let's correct that. Let's start over.

Wait, the original inequality is \(-5\sqrt{x - 2} < -10\) (I think there was a typo, the square root is of \(x - 2\), not \(x\) minus \(2\) outside). Let's assume it's \(-5\sqrt{x - 2} < -10\).

So, starting over with \(-5\sqrt{x - 2} < -10\).

Step1: Divide both sides by -5 (reverse inequality)

Divide both sides by \(-5\). Since we divide by a negative number, the inequality sign flips:
\[
\sqrt{x - 2}>\frac{-10}{-5}=2
\]

Step2: Square both sides (note domain)

Now, square both sides of the inequality \(\sqrt{x - 2}>2\). Since both sides are non - negative (the square root is non - negative and \(2>0\)), we can square both sides without changing the inequality direction:
\[
x - 2>2^{2}=4
\]

Step3: Solve for x

Add \(2\) to both sides of the inequality \(x - 2>4\):
\[
x>4 + 2=6
\]
Also, we need to consider the domain of the square root function. For \(\sqrt{x - 2}\) to be defined, \(x-2\geq0\), i.e., \(x\geq2\). But our solution from the inequality is \(x > 6\), which is within the domain \(x\geq2\).

To graph the solution: On a number line, we mark the point \(x = 6\) with an open circle (because the inequality is \(x>6\), not \(x\geq6\)) and then shade the region to the right of \(6\) to represent all real numbers greater than \(6\).

Answer:

The solution to the inequality is \(x > 6\), and on the number line, we draw an open circle at \(6\) and shade to the right.