To solve this inequality, we first need to find the zeros of the expression inside the logarithm.
Given: log(1/2)(x^2 + 3x - 4) <= 0
First, factor the quadratic expression inside the logarithm: (x^2 + 3x - 4) = (x + 4)(x - 1)
Now we have: log(1/2)(x + 4)(x - 1) <= 0
Next, we need to find the values of x that make the expression inside the logarithm equal to zero: (x + 4)(x - 1) = 0 x + 4 = 0 or x - 1 = 0 x = -4 or x = 1
Now, we have three intervals to test: 1) x < -4 2) -4 < x < 1 3) x > 1
1) Let's test for x < -4: Let x = -5: log(1/2)((-5 + 4)(-5 - 1)) = log(1/2)(-1 * -6) = log(1/2)(6) > 0
2) Now let's test for -4 < x < 1: Let x = 0: log(1/2)((0 + 4)(0 - 1)) = log(1/2)(4 * -1) = log(1/2)(-4) ≤ 0
3) Finally, test for x > 1: Let x = 2: log(1/2)((2 + 4)(2 - 1)) = log(1/2)(6 * 1) = log(1/2)(6) > 0
Therefore, the solution to the inequality log(1/2)(x^2 + 3x - 4) ≤ 0 is: x ∈ (-4, 1]
To solve this inequality, we first need to find the zeros of the expression inside the logarithm.
Given: log(1/2)(x^2 + 3x - 4) <= 0
First, factor the quadratic expression inside the logarithm:
(x^2 + 3x - 4) = (x + 4)(x - 1)
Now we have: log(1/2)(x + 4)(x - 1) <= 0
Next, we need to find the values of x that make the expression inside the logarithm equal to zero:
(x + 4)(x - 1) = 0
x + 4 = 0 or x - 1 = 0
x = -4 or x = 1
Now, we have three intervals to test:
1) x < -4
2) -4 < x < 1
3) x > 1
1) Let's test for x < -4:
Let x = -5:
log(1/2)((-5 + 4)(-5 - 1)) = log(1/2)(-1 * -6) = log(1/2)(6) > 0
2) Now let's test for -4 < x < 1:
Let x = 0:
log(1/2)((0 + 4)(0 - 1)) = log(1/2)(4 * -1) = log(1/2)(-4) ≤ 0
3) Finally, test for x > 1:
Let x = 2:
log(1/2)((2 + 4)(2 - 1)) = log(1/2)(6 * 1) = log(1/2)(6) > 0
Therefore, the solution to the inequality log(1/2)(x^2 + 3x - 4) ≤ 0 is: x ∈ (-4, 1]