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

1 Answer
KillerBunny
Oct 13, 2015

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.

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