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Problem 544

# Solve the inequality $$\mathrm{x}^{2}>4$$

Expert verified
The inequality $$x^2 > 4$$ is satisfied when $$x < -2$$ or $$x > 2$$. The solution is $$x \in (-\infty, -2) \cup (2, \infty)$$.
See the step by step solution

## Step 1: Identify the related equation

We will first consider the related equation $$x^2 = 4$$. Solving this equation will help us identify the critical points where the inequality changes from being true to false.

## Step 2: Solve the related equation

To solve the equation $$x^2 = 4$$, we take the square root of both sides of the equation: $$\sqrt{x^2} = \sqrt{4}$$ This gives us two possible solutions: $$x = 2$$ and $$x = -2$$. These critical points will help us determine the intervals that satisfy the inequality $$x^2 > 4$$.

## Step 3: Test the intervals on the number line

Now that we have the critical points, we will test the intervals created by them on the number line. We will check the inequality for a value in each interval: 1. Interval $$-\infty < x < -2$$: We can test $$x = -3$$, which gives $$(-3)^2 > 4 \Rightarrow 9 > 4$$, which is true. 2. Interval $$-2 < x < 2$$: We can test $$x = 0$$, which gives $$0^2 > 4 \Rightarrow 0 > 4$$, which is false. 3. Interval $$2 < x < \infty$$: We can test $$x = 3$$, which gives $$(3)^2 > 4 \Rightarrow 9 > 4$$, which is true.

## Step 4: Write the intervals that satisfy the inequality

From the tests in Step 3, we can see that the inequality $$x^2 > 4$$ is satisfied when $$x < -2$$ or $$x > 2$$. So, we can write the solution as the union of these two intervals: $$x \in (-\infty, -2) \cup (2, \infty)$$

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