Solve the inequality. Express your solution using interval notation and please show all of your work.
x^4 - 7x^2 - 14 > 0
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To solve the inequality x^4 - 7x^2 - 14 > 0, we need to find the values of x that make the expression on the left side of the inequality greater than zero. Here's how we can do it:
Step 1: Factorize the expression if possible.
Unfortunately, the expression x^4 - 7x^2 - 14 cannot be easily factorized. So we need an alternative method.
Step 2: Substitute a variable to help solve the inequality.
We can substitute another variable to simplify the inequality. Let's use y = x^2. Now, the inequality becomes y^2 - 7y - 14 > 0.
Step 3: Solve the quadratic inequality.
To solve the quadratic inequality y^2 - 7y - 14 > 0, we need to find the values of y that make the expression greater than zero.
Step 3.1: Factorize the quadratic expression.
The expression y^2 - 7y - 14 can be factorized as (y - 2)(y - 7). So we have (y - 2)(y - 7) > 0.
Step 3.2: Determine the sign of each factor.
Let's analyze the sign of each factor:
- When y - 2 > 0, we have y > 2.
- When y - 7 > 0, we have y > 7.
Step 3.3: Determine the sign of the inequality.
Next, let's determine the intervals where the inequality (y - 2)(y - 7) > 0 is satisfied. We can do this by using a number line.
On the number line, we mark the values of y where the factors change sign.
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2 7
Step 3.4: Analyze the intervals.
There are three intervals: (-∞, 2), (2, 7), and (7, +∞).
- For the interval (-∞, 2), both factors are negative. So (y - 2)(y - 7) < 0.
- For the interval (2, 7), the factor (y - 2) is positive while (y - 7) is negative. So (y - 2)(y - 7) > 0.
- For the interval (7, +∞), both factors are positive. So (y - 2)(y - 7) < 0.
Step 4: Convert back to the original variable, x.
Now that we have the intervals for y, let's convert them back to x using the substitution y = x^2.
- For the interval (-∞, 2), we substitute y = x^2 and solve x^2 < 2. This means -√2 < x < √2.
- For the interval (2, 7), we substitute y = x^2 and solve 2 < x^2 < 7. This means -√7 < x < -√2 or √2 < x < √7.
- For the interval (7, +∞), we substitute y = x^2 and solve x^2 > 7. This means x < -√7 or x > √7.
Step 5: Express the solution in interval notation.
Combining all the intervals, the solution to the inequality x^4 - 7x^2 - 14 > 0 in interval notation is:
(-∞, -√7) U (-√2, -√7) U (-√2, √2) U (√2, √7) U (√7, +∞)