Inequalities question

[bumblebee123]

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]

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

 

[Harry_the_cat]

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]

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!

 

[Steven G]

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.