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

Oct 13, 2015

The inequality holds for $x \setminus \le - 2$ and $\frac{1}{2} \setminus \le x \setminus \le 4$.

#### 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, $2 x - 1$ is positive if and only if $x > \frac{1}{2}$, $x + 2$ is positive if and only if $x > - 2$, and $x - 4$ is positive if and only if $x > 4$.

So, the important points are $- 2$, $\frac{1}{2}$ and $4$.

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

Between $- 2$ and $\frac{1}{2}$, $x + 2$ is positive and the other two are negative, so the product is positive.

Between $\frac{1}{2}$ 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.