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 $\displaystyle x \ge 0$, then $\displaystyle |7x| = 7x$ and $\displaystyle 7x = 4-x \Rightarrow x = \frac{1}{2}$.
If $\displaystyle x \le 0$ then $\displaystyle |7x| = -7x$, $\displaystyle -7x = 4-x \Rightarrow x = -\frac{2}{3}$. Both of these solutions satisfy the original equation.
2. Factor to $\displaystyle x(x-3)(x-5)(x+6) \le 0$. Exactly one or three of these factors must be positive. Use casework.
We know that $\displaystyle \displaystyle \begin{align*} x(x - 3)(x - 5)(x + 6) \leq 0 \end{align*}$. The LHS will be equal to 0 when $\displaystyle \displaystyle \begin{align*} x = \{ -6, 0, 3, 5 \} \end{align*}$. 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.
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 $\displaystyle x(x-3)(x-5)(x-6)$ is positive. However we are only interested in the intervals where LHS is negative. Hence $\displaystyle x \le 0$ is not part of the solution set.
Repeat the same for $\displaystyle 0 < x < 3$, $\displaystyle 3 < x < 5$, $\displaystyle 5 < x < 6$, and $\displaystyle 6 < x$.
I thought I explained this...
There are five important intervals you need to test, namely all the intervals around the roots, so $\displaystyle \displaystyle \begin{align*} x < -6, -6 < x < 0, 0 < x < 3, 3 < x < 5, x> 5 \end{align*}$.
So choose a value of x in the first interval, say x = -7. Then we have
$\displaystyle \displaystyle \begin{align*} x(x - 3)(x - 5)(x + 6) &= -7(-7 - 3)(-7 - 5)(-7 + 6) \\ &= -7(-10)(-12)(-1) \\ &= 840 > 0 \end{align*}$
Since this value is positive, then we can say that the region in which x < -6 does not satisfy $\displaystyle \displaystyle \begin{align*} x(x - 3)(x - 5)(x + 6) \leq 0 \end{align*}$.
Now try testing values of x from each of the other important regions. Accept the regions which give negative values for the function.
I've always found that drawing a graph makes solving these inequalities easier.
The equation is: $\displaystyle x(x-3)(x-5)(x-6) \le 0$.
Go here: x(x-3)(x-5)(x-6) - 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 $\displaystyle x(x-3)(x-5)(x-6) $ is negative when $\displaystyle 0 \le x \le3$
Do you see how I got that? Now you find the next solution