solve the inequality: x-3/x-2 + 12/x^2-x-2> 8/X+1
multiply both sides by (x-2)(x+1)
(x-3)(x+1)+12>8
x^2-2x-3+4>0
(x-1)^2>0
which means that x can be any value except 1
Hmmm. I get
(x-3)/(x-2) + 12/(x^2-x-2) > 8/(x+1)
(x-3)(x+1) + 12 > 8(x-2)
x^2-2x-3+12 > 8x-16
x^2-10x+25 > 0
(x-5)^2 > 0
So, all x except x = 2,5 and -1
Oddly enough, it appears that the real solution is x < -1 or x > 2 and x not 5. See
http://www.wolframalpha.com/input/?i=%28x-3%29%2F%28x-2%29+%2B+12%2F%28x^2-x-2%29+%3E+8%2F%28x%2B1%29+for+-2+%3Cx+%3C+6
Maybe you can spot my math error. Probably involves clearing the fractions, since the direction of the inequality will change with negative values.
To solve the given inequality:
1. Determine the restricted values. Set the denominators equal to zero and solve for x.
(x - 2)(x + 1)(x - 1) = 0
The restricted values are: x = -1, 1, 2.
2. Find the sign chart. Divide the number line into four intervals based on the restricted values:
(-∞, -1), (-1, 1), (1, 2), (2, ∞).
3. Test a value from each interval. Choose a number within each interval and substitute it into the inequality to determine the sign.
For example:
Test x = -2:
Substitute x = -2 into the inequality:
(-2 - 3)/(-2 - 2) + 12/((-2)^2 - (-2) - 2) > 8/(-2 + 1)
-5/(-4) + 12/8 > 8/-1
5/4 + 3/2 > -8
Test x = 0 and x = 1/2 similarly to get the sign for the remaining intervals.
4. Construct the sign chart. Based on the tested values, determine the sign for each interval.
Interval | Test Value | Sign
(-∞, -1) | -2 | >
(-1, 1) | 0 | <
(1, 2) | 1/2 | >
(2, ∞) | 3 | >
5. Determine the solution. The solution is obtained by analyzing the sign chart.
- The inequality is true for the intervals where the sign is ">", so the solution intervals are (-∞, -1) and (1, 2).
- The inequality is false for the intervals where the sign is "<", so the solution intervals are not considered.
6. Write the solution set. Combine the intervals to get the final solution.
Solution: (-∞, -1) ∪ (1, 2)
To solve the given inequality, we will follow these steps:
Step 1: Simplify the expression.
Step 2: Find the common denominator.
Step 3: Combine the fractions.
Step 4: Get rid of the denominators.
Step 5: Arrange all terms on one side of the inequality.
Step 6: Factorize the resulting quadratic inequality.
Step 7: Find the critical values.
Step 8: Test the intervals between the critical values.
Step 9: Determine the solution.
Now, let's go through each step in detail:
Step 1: Simplify the expression.
The given inequality is:
(x - 3) / (x - 2) + 12 / (x^2 - x - 2) > 8 / (x + 1)
Step 2: Find the common denominator.
The common denominator is (x - 2)(x + 1).
Step 3: Combine the fractions.
The expression can be rewritten as follows:
[(x - 3)(x + 1) + 12] / [(x - 2)(x + 1)] > 8 / (x + 1)
Step 4: Get rid of the denominators.
Multiply both sides of the inequality by (x - 2)(x + 1) to eliminate the denominators:
[(x - 3)(x + 1) + 12] > 8
Step 5: Arrange all terms on one side of the inequality.
Expand and simplify the left side of the inequality:
(x^2 - 2x - 3 + x - 3 + 12) > 8
(x^2 - x + 6) > 8
Step 6: Factorize the resulting quadratic inequality.
To solve the quadratic inequality, factorize it:
(x - 2)(x + 3) > 0
Step 7: Find the critical values.
The critical values are the values of x where the expression equals zero.
Setting each factor equal to zero:
x - 2 = 0 -> x = 2
x + 3 = 0 -> x = -3
The critical values are x = 2 and x = -3.
Step 8: Test the intervals between the critical values.
To test the intervals, choose a point within each interval and substitute it into the original inequality. If the inequality is satisfied, that interval is part of the solution.
Testing x = -4 (any value less than -3):
(x - 2)(x + 3) = (-4 - 2)(-4 + 3) = (-6)(-1) = 6 > 0
Since 6 > 0, the interval (-∞, -3) is part of the solution.
Testing x = 0 (any value between -3 and 2):
(x - 2)(x + 3) = (0 - 2)(0 + 3) = (-2)(3) = -6 < 0
Since -6 < 0, the interval (-3, 2) is not part of the solution.
Testing x = 3 (any value greater than 2):
(x - 2)(x + 3) = (3 - 2)(3 + 3) = (1)(6) = 6 > 0
Since 6 > 0, the interval (2, ∞) is part of the solution.
Step 9: Determine the solution.
From the test intervals, we conclude that the solution to the inequality is:
x ∈ (-∞, -3) U (2, ∞)