Thread: solution set

can anyone tell me what solution i could use to find the solution set of:

1.) |7x| = 4 - x

2. (x^2 - 8x + 15) ( x^2 + 6x) <= 0


thank you

1. If , then and .

If then , . Both of these solutions satisfy the original equation.


2. Factor to . Exactly one or three of these factors must be positive. Use casework.

by cases : is it possible to say that (x-3)>= 0 , or (x-5)> = 0, or (x+6)>= 0, what about x ? is that x > =0
is that how i am going to solve for the values of x or roots sir?

thank you

Originally Posted by rcs

by cases : is it possible to say that (x-3)>= 0 , or (x-5)> = 0, or (x+6)>= 0, what about x ? is that x > =0
is that how i am going to solve for the values of x or roots sir?

thank you

We know that . The LHS will be equal to 0 when . The only times that a function will go from negative to positive or vice versa is when it crosses the x axis, so at these values of x. So test values of x on each side of these intercepts to see where the function is negative.

will this have a very long solution? to find the solution set ?

No, it's not long at all. There's a little brute-force involved, but not that much.

thank you Guys... im trying but still i cannot figure out

You want to solve . You know that if x = 0, 3, 5, or 6 you obtain equality. If , is the LHS positive or negative?

do you mean sir that i have to make this - x <= 0 then x <= o : -x - 3<= 0 ; - x - 5 <= 0 : - x - 6 <= 0 .... so solve for x?

Sort of, but you're not solving for x. If x is negative, then x-3, x-5, and x-6 are negative, and the product is positive. However we are only interested in the intervals where LHS is negative. Hence is not part of the solution set.


Repeat the same for , , , and .

huhuhuhuh what can i do sir ... my brain is not working anymore please help

I thought I explained this...

There are five important intervals you need to test, namely all the intervals around the roots, so .

So choose a value of x in the first interval, say x = -7. Then we have



Since this value is positive, then we can say that the region in which x < -6 does not satisfy .

Now try testing values of x from each of the other important regions. Accept the regions which give negative values for the function.

Originally Posted by rcs

do you mean sir that i have to make this - x <= 0 then x <= o : -x - 3<= 0 ; - x - 5 <= 0 : - x - 6 <= 0 .... so solve for x?

I've always found that drawing a graph makes solving these inequalities easier.

The equation is: .

Go here: x&#40;x-3&#41;&#40;x-5&#41;&#40;x-6&#41; - Wolfram|Alpha to look at your polynomial.
Now found the values of x that make the graph go below the x-axis - as that's essentially what the equation is saying.

So looking at the graph, it's obvious that values of is negative when

Do you see how I got that? Now you find the next solution