Inequalities question

bumblebee123

Junior Member
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Jan 3, 2018
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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 :)
 
(x2)(x+3)0 is true only when (x2)0 and (x+3)0\displaystyle (x-2)(x+3)\leq 0 \text{ is true only when }(x-2)\leq 0 \text{ and }(x+3)\geq 0
(,2][3,)=[3,2]\displaystyle (-\infty,2] \cap [-3,\infty) = [-3,2]
 
One way of looking at this type of question is to consider the graph of y=x2+x6=(x2)(x+3)\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 y=x2+x6\displaystyle y=x^2+x-6 lies below (or on) the x-axis for 3x2\displaystyle -3\leq x \leq 2.

This is when x2+x60\displaystyle x^2 + x - 6 \leq 0
 
One way of looking at this type of question is to consider the graph of y=x2+x6=(x2)(x+3)\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 y=x2+x6\displaystyle y=x^2+x-6 lies below (or on) the x-axis for 3x2\displaystyle -3\leq x \leq 2.

This is when x2+x60\displaystyle x^2 + x - 6 \leq 0

ah! okay. this makes a lot of sense. thank you so much! :)
 
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.
 
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