Get the complete, step-by-step math solution for: "Solve the following inequation : (x - 2)/(x + 5) > 2.". Powered by SolveForX AI math tutor.
Step 2: Combine terms into a single fraction
Next, we combine the terms on the left side into a single fraction by finding a common denominator, which is x+5. We then simplify the numerator.
Step 4: Multiply by -1 and reverse inequality
To make the leading coefficient of x in the numerator positive, we multiply both the numerator and the right side of the inequality by −1. Remember to reverse the inequality sign when multiplying or dividing by a negative number.
Step 5: Determine critical points and intervals
The critical points are the values of x that make the numerator or the denominator equal to zero. These points divide the number line into intervals where the expression's sign can be analyzed. The critical points are x=−12 (from x+12=0) and x=−5 (from x+5=0).
Step 6: Analyze the sign of the expression
We test values in the intervals (−∞,−12), (−12,−5), and (−5,∞).
For x<−12 (e.g., x=−13): −13+5−13+12=−8−1=81>0.
For −12<x<−5 (e.g., x=−10): −10+5−10+12=−52<0.
For x>−5 (e.g., x=0): 0+50+12=512>0.
We are looking for where x+5x+12<0, which is the interval (−12,−5).