Solve the inequality. Express your solution using notation and please show all of your work.
x^4<16x^2
x^4 < 16x^2
x^2(x^2 - 16) < 0
x^2 is always positive, so
x^2 < 16
so, for -4 < x < 4
x^4 < 16x^2
at x=0, x^4 = 16x^2, so 0 is excluded from the solution
To solve the inequality x^4 < 16x^2, we can start by subtracting 16x^2 from both sides:
x^4 - 16x^2 < 0
Now, we can factor the left side of the inequality:
x^2(x^2 - 16) < 0
Next, we can factor the difference of squares within the parentheses:
x^2(x + 4)(x - 4) < 0
Now, we have three factors: x^2, (x + 4), and (x - 4). To determine the sign of the expression, we can analyze each factor separately.
For x^2, the sign is determined by the value of x^2. Since x^2 is always non-negative (positive or zero), the factor x^2 is positive or zero for all values of x.
For (x + 4), the sign is determined by the value of x + 4. The factor is positive for x > -4 and negative for x < -4.
For (x - 4), the sign is determined by the value of x - 4. The factor is positive for x > 4 and negative for x < 4.
To determine the signs of the entire expression, we can create a sign chart, considering the intervals defined by the zeros of each factor (-4 and 4):
-∞ -4 4 +∞
x^2 + + + +
(x + 4) - - + +
(x - 4) - + + +
To determine the sign of the inequality, we need to look at the product of all three factors.
- If the product is negative, x^4 - 16x^2 < 0.
- If the product is positive, x^4 - 16x^2 > 0.
From the sign chart, we can see that the expression is negative for -4 < x < 4, and positive for x < -4 or x > 4.
Therefore, the solution to the inequality x^4 < 16x^2 is:
x < -4 or -4 < x < 4 or x > 4