Obviously, there are 3 critical points to consider: x = -3, x = 1 and x = 5. At these points, the function is 0, so evaluating at these points is not very useful, so look at points very near them. For instance, let's start with the point 1.01. The first term is positive, the second term is positive, but the third term is negative - so the function is negative and satisfies the inequality. You get one piece of information: x>1 What about at -3.01? x+3 < 0, x-1 < 0, x - 5 < 0. Three negatives still makes your function negative, so x < -3 is another piece. Now what about 5.01? x+3 > 0, x-1 > 0, x-5 > 0. Three positives, so your function is positive and doesn't satisfy the equation. So x can't be more than 5, it must be x < 5. Now you have three pieces. x < -3, x > 1, x < 5. Put them together to get: x < -3, 1 < x < 5.
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-2 - x ≥ -1
-x ≥ 1
x ≤ -1
{x: x ≤ -1}
Obviously, there are 3 critical points to consider: x = -3, x = 1 and x = 5. At these points, the function is 0, so evaluating at these points is not very useful, so look at points very near them. For instance, let's start with the point 1.01. The first term is positive, the second term is positive, but the third term is negative - so the function is negative and satisfies the inequality. You get one piece of information: x>1 What about at -3.01? x+3 < 0, x-1 < 0, x - 5 < 0. Three negatives still makes your function negative, so x < -3 is another piece. Now what about 5.01? x+3 > 0, x-1 > 0, x-5 > 0. Three positives, so your function is positive and doesn't satisfy the equation. So x can't be more than 5, it must be x < 5. Now you have three pieces. x < -3, x > 1, x < 5. Put them together to get: x < -3, 1 < x < 5.