Write two possible quartic inequalities, one using the less than or equal to symbol (≤) and the other using the greater than or equal to symbol (≥), that correspond to the following solution : -6-√2 <x< -6+√2 or 6-√2 <x< 6+√2
1) -(x+6)^4 + 2 ≤ 0
2) (x-6)^4 - 2 ≥ 0
Sure! Here are two quartic inequalities that correspond to the given solution:
1. Using the less than or equal to symbol (≤):
(x + 6 - √2)(x + 6 + √2)(x - 6 + √2)(x - 6 - √2) ≤ 0
2. Using the greater than or equal to symbol (≥):
(x + 6 - √2)(x + 6 + √2)(x - 6 + √2)(x - 6 - √2) ≥ 0
These inequalities are formed by multiplying out the factors corresponding to the given solution and setting them less than or equal to 0 in the first case, and greater than or equal to 0 in the second case.
To find the quartic inequalities corresponding to the given solution, we need to consider the interval notation of the solution and convert it into inequality form.
The given solution is: -6 - √2 < x < -6 + √2 or 6 - √2 < x < 6 + √2.
1. Less than or equal to inequality (≤):
For the lower limit, -6 - √2 < x, we use the less than or equal to symbol (≤). This means x is either less than or equal to -6 - √2. Similarly, for the upper limit, x < -6 + √2.
Combining both inequalities, we get: -6 - √2 ≤ x < -6 + √2.
2. Greater than or equal to inequality (≥):
For the lower limit, -6 - √2 < x, we use the greater than or equal to symbol (≥). This means x is either greater than or equal to -6 - √2. Similarly, for the upper limit, x < 6 + √2.
Combining both inequalities, we get: -6 - √2 ≥ x > 6 + √2.
So, the two possible quartic inequalities corresponding to the given solution are:
1. -6 - √2 ≤ x < -6 + √2 (using the less than or equal to symbol)
2. -6 - √2 ≥ x > 6 + √2 (using the greater than or equal to symbol)
-6-√2 <x< -6+√2
-√2 < x+6 < √2
(x+6)^2 < 2
x^2 + 12x + 36 < 2
x^2 + 12x + 34 < 0