solve the inequality. Express the solution using interval notation and graph the solution set on the real number line
x-4/x^2-4 greater or equal to O
(x-4)/(x^2-4) >= 0
Either both top and bottom are positive, or both negative
x-4 > 0 if x > 4
x^2-4 > 0 if |x| > 2
only x > 4 satisfies both requirements
x-4 < 0 if x < 4
x^2 - 4 < 0 if |x| < 2
So, -2 < x < 2
Solution set: (-2,2)U(4,oo)
To solve the inequality (x - 4)/(x^2 - 4) ≥ 0, we need to find the values of x that satisfy this inequality.
Step 1: Determine the critical points.
The critical points are the values of x that make the numerator or denominator equal to zero. In this case, the critical points occur when x - 4 = 0 or x^2 - 4 = 0.
From x - 4 = 0, we find x = 4.
From x^2 - 4 = 0, we can factor the equation into (x - 2)(x + 2) = 0. Thus, x = -2 or x = 2.
Step 2: Determine the sign of the inequality in each interval.
To do this, we choose test points within each interval and evaluate the sign of the expression (x - 4)/(x^2 - 4).
- For x < -2, we choose a test point such as x = -3. Plugging it into the expression, we get (-3 - 4)/((-3)^2 - 4) = -7/5. Since this is negative, the inequality is not satisfied in this interval.
- For -2 < x < 2, we choose a test point such as x = 0. Plugging it into the expression, we get (0 - 4)/(0^2 - 4) = -1. Since this is negative, the inequality is satisfied in this interval.
- For x > 2, we choose a test point such as x = 3. Plugging it into the expression, we get (3 - 4)/(3^2 - 4) = -1/5. Since this is negative, the inequality is not satisfied in this interval.
Step 3: Express the solution in interval notation and graph it on the number line.
Based on the sign analysis, the solution set consists of the interval (-2, 2]. The square brackets indicate that the value of 2 is included in the solution set.
On the number line, we can represent the solution as follows:
-inf -2 2 4 +inf
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