|(x-3)/(x+5)|< 2
i've asked so many ppl, but they all give me different answers :(
Well if the expression is less than two, and the abs function is positive, then the ratio is between 0 and 2. Lets look inside the abs function, it can be negative, because the abs will make it positive.
If that is so, then...
-2 < (x-3)/(x+5)<2 multiply the expression by x+5
-2 (x+5)< (x-3) <2(x+5)
-2x -10 < x-3 < 2x +10
-10 < x-3+ 2x < 4x +10
Now, examining this part -10 < x-3+ 2x
or -7 <3x
or x > -7/3
x>-2.33333 Examining the other part.. x-3+ 2x < 4x +10
-3 < 3x +10
-13 < 3x
x > -13/3 x>- 4.3333 but both of these must occur, so x> -2.333..
Now to check it. Let x=-2, 4, and -5(it should not check on this one. Let me know if you have difficulty.
To solve the inequality |(x-3)/(x+5)| < 2, you followed the correct steps so far.
Here's a step-by-step explanation of the solution:
1. Start with the given inequality: |(x-3)/(x+5)| < 2.
2. Since the absolute value function can be positive or negative, split the inequality into two cases:
Case 1: (x-3)/(x+5) > -2 (no change because the absolute value is positive)
Case 2: (x-3)/(x+5) < 2 (change the direction of the inequality because the absolute value is negative)
3. Focus on Case 1: (x-3)/(x+5) > -2.
Multiply both sides of the inequality by (x+5) to remove the denominator:
(x+5) * (x-3)/(x+5) > -2 * (x+5)
Cancel out (x+5) from the left side:
(x-3) > -2 * (x+5)
4. Simplify the expression: (x-3) > -2x - 10.
Expand -2 * (x+5):
(x-3) > -2x - 10.
5. Rearrange the inequality: 3x + 7 > 0.
Add 2x to both sides:
3x + 2x + 7 > 0.
Combine like terms:
5x + 7 > 0.
6. Solve for x: x > -7/5.
7. Now, move on to Case 2: (x-3)/(x+5) < 2.
Repeat the steps 3 to 6 with the direction of the inequality changed.
8. Simplify the expression: (x-3) < 2x + 10.
Expand 2 * (x+5):
(x-3) < 2x + 10.
9. Rearrange the inequality: -3x - 13 < 0.
Subtract 2x from both sides:
-3x - 2x - 13 < 0.
Combine like terms:
-5x - 13 < 0.
10. Solve for x: x > -13/5.
11. Now, we need to find the intersecting values from both cases.
12. The intersecting values of x are x > -7/5 and x > -13/5.
13. To check the solution, substitute values of x into the original inequality:
- Let's pick x = -2:
|(-2-3)/(-2+5)| = |-5/3| = 5/3 which is indeed less than 2.
- Let's pick x = 4:
|(4-3)/(4+5)| = |1/9| = 1/9 which is less than 2.
- Let's pick x = -5:
|(-5-3)/(-5+5)| = |-8/0| = undefined, so -5 is not a valid solution.
14. The valid solution is x > -7/5.
Therefore, the solution to the inequality |(x-3)/(x+5)| < 2 is x > -7/5.