Solve the following inequality.
(x-4/-x+2)>0
Write your answer using interval notation.
You need parentheses.
x - (4/-x) + 2 > 0 ?
(x - 4)/(-x + 2) > 0 ?
Please, re-state the problem with parentheses.
To solve the inequality (x-4)/(-x+2)>0, we need to find the values of x that make the expression greater than zero.
To do this, we first need to find the values of x that make the expression equal to zero. This occurs when the numerator, (x-4), equals zero. So, setting (x-4) = 0 and solving for x, we find x = 4.
Next, we need to find the values of x that make the denominator, (-x+2), equal to zero. Setting (-x+2) = 0 and solving for x, we find x = 2.
Now, we have three critical points: x = 4, x = 2 from setting the numerator and denominator to zero, and -∞ from the negative sign in the denominator.
To determine the intervals where the expression is greater than zero, we can use a number line.
Let's test the intervals one by one:
1. For x < 2:
- If we choose x = 0 (a value less than 2), the expression becomes (-4 / -2) = 2, which is positive. So, this interval satisfies the inequality.
2. For 2 < x < 4:
- If we choose x = 3 (a value between 2 and 4), the expression becomes (-1 / -1) = 1, which is not greater than zero. So, this interval does not satisfy the inequality.
3. For x > 4:
- If we choose x = 5 (a value greater than 4), the expression becomes (1 / -3) = -1/3, which is negative. So, this interval does not satisfy the inequality.
Therefore, the solution to the inequality (x-4)/(-x+2) > 0 is x < 2 or x > 4.
In interval notation, the solution can be written as (-∞, 2) ∪ (4, +∞).