Question

How do you solve `(2x-1)(x+2)(x-4)<=0`?

Answer 1

The inequality holds for `x\le -2` and `1/2\le x \le 4`.

Explanation:

Keep in mind the rule for multiplying:

• Positive times positive is positive;
• Positive times negative is negative;
• Negative times positive is negative;
• Negative times negative is positive.

Combining these rules, you can clearly see that a product of factors is negative if and only if there is an odd number of negative factors (in this case, one or all three).

Now, `2x-1` is positive if and only if `x>1/2`, `x+2` is positive if and only if `x> -2`, and `x-4` is positive if and only if `x>4`.

So, the important points are `-2`, `1/2` and `4`.

Before `-2`, all three factors are negative, so the product is negative.

Between `-2` and `1/2`, `x+2` is positive and the other two are negative, so the product is positive.

Between `1/2` and `4`, only `x-4` is negative, and the other two are positive, so the product is negative.

After `4`, all the factors are positive, so the product is positive.