Solve the following inequality. Write the answer in interval notation.
x^4>4x^2
x^4>4x^2
x^4 - 4x^2 > 0
x^2(x^2 - 4) > 0
x^2(x+2)(x-2) > 0
critical values are -2, 0, +2
for x < -2 , --->(+)(-)(-) >0 , so true
for x = 0 , no good
for -2 < x <0 , say x = -1
---> (+)(+)(-) < 0 , false
for 0 < x < 2 , ---> (+)(+)(-) < 0 , false
for x > 2, (+)(+)(+) > 0 , so true
solution:
x < -2 OR x > 2
check with Wolfram, see where the graph lies above the x axis.
http://www.wolframalpha.com/input/?i=plot+x%5E4+-+4x%5E2
To solve the inequality x^4 > 4x^2, we can start by subtracting 4x^2 from both sides to bring everything to one side of the inequality:
x^4 - 4x^2 > 0
Next, we can factor out the common term, x^2:
x^2(x^2 - 4) > 0
Now, we have a product of two factors. To determine the values of x that satisfy the inequality, we need to consider two cases:
Case 1: x^2 > 0
If x^2 > 0, it means that x^2 - 4 can be either positive or negative. However, since a positive number multiplied by a positive number is always positive, we know that the product of these two factors will be positive. So, all values of x that satisfy x^2 > 0 will also satisfy the inequality.
Case 2: x^2 - 4 > 0
To find the values of x that satisfy this part of the inequality, we solve the quadratic inequality:
x^2 - 4 > 0
(x - 2)(x + 2) > 0
Now, we can use the concept of sign charts or a number line to determine the intervals where the inequality holds true.
1) For x < -2, both factors (x - 2) and (x + 2) are negative, which results in a positive product. Therefore, this interval satisfies the inequality.
2) For -2 < x < 2, (x - 2) is negative, while (x + 2) is positive. Multiplying a negative number by a positive number gives a negative product. Hence, this interval does not satisfy the inequality.
3) For x > 2, both factors are positive, resulting in a positive product. So, this interval satisfies the inequality.
Combining the results from both cases, we find that the solution to the inequality x^4 > 4x^2 in interval notation is:
(-∞, -2) U (2, ∞)