1. x( x-1)(x+2)>0 (quadratic Inequality)
2. 2x-3/x+1 ≤ 1 (inequality involving quotient)
First we want to get everything onto one side of the equation. That's already done here.
Next we want to identify "critical points." This is where the LHS of the equation is 0, or where the denominator is 0. Here we have no denominator, so our critical points are x = -2, 0, 1.
Now we want to break the real line into pieces, given by the critical points. In this case we have 4 intervals:
Now we want to test values of x from each interval in the inequality to see which ones give the correct result:
Only two of these intervals have x values that work in the inequality. These intervals are the solution set. Thus the answer is .
The answer to the second problem is . (The 4 is included in the solution set because we have a symbol. Why isn't -1 included in the solution set?)
The way I learned how to graph the first problem (there are at least two conventions being used)
would be to put an open circle on x = -2, x = 0, and x = 1, then draw a line connecting x = -2 and x = 0 to indicate that these points are not part of the solution set, then draw a line to the right starting at x = 1 and continuing to . (I put an arrow on the line from x = 1.)
For the second problem:
I put an open circle on -1 and a closed circle on 4 (to indicate that x = 4 is part of the solution) and connect x = -1 and x = 4.
Why can't x = -1 in the second solution? First of all, PLEASE use parenthesis!!
What you wanted was "(x - 4)/(x + 1)"
Anyway, plug x = -1 into the original inequality. Is this possible? What about x = 4?