# Inequalities question

#### bumblebee123

##### Junior Member
can anyone help to explain the answer to this question?

question: solve the inequality x^2 + x - 6 ≤ 0

I can begin to solve this: ( x - 2 ) ( x + 3 ) ≤ 0

x ≤ 2 which is correct

x ≤ - 3 which is not correct

the answer is - 3 x

why does the inequality sign change?

any help would be really appreciated #### Romsek

##### Full Member
$$\displaystyle (x-2)(x+3)\leq 0 \text{ is true only when }(x-2)\leq 0 \text{ and }(x+3)\geq 0$$
$$\displaystyle (-\infty,2] \cap [-3,\infty) = [-3,2]$$

#### Harry_the_cat

##### Senior Member
One way of looking at this type of question is to consider the graph of $$\displaystyle y= x^2+x -6 =(x-2)(x+3)$$

This is a U shaped parabola with x-intercepts of -3 and 2. Quickly sketch the graph

You can then see that the graph of $$\displaystyle y=x^2+x-6$$ lies below (or on) the x-axis for $$\displaystyle -3\leq x \leq 2$$.

This is when $$\displaystyle x^2 + x - 6 \leq 0$$

#### bumblebee123

##### Junior Member
One way of looking at this type of question is to consider the graph of $$\displaystyle y= x^2+x -6 =(x-2)(x+3)$$

This is a U shaped parabola with x-intercepts of -3 and 2. Quickly sketch the graph

You can then see that the graph of $$\displaystyle y=x^2+x-6$$ lies below (or on) the x-axis for $$\displaystyle -3\leq x \leq 2$$.

This is when $$\displaystyle x^2 + x - 6 \leq 0$$
ah! okay. this makes a lot of sense. thank you so much! #### Jomo

##### Elite Member
ah! okay. this makes a lot of sense. thank you so much! Or you can consider that a product of two factors are negative when they have different signs, ie +*- = - OR -*+ = -

So you want (x-2)>0 AND (x+3)< 0 OR (x-2)<0 AND (x+3)> 0
You need to use this method if you do not have a quadratic equation.

• Harry_the_cat