+0

Help with inequalities

0
388
4

Find all solutions to the inequality $$\frac{(2x-7)(x-3)}{x} \ge 0$$ in interval notation

Aug 7, 2019

#1
+6203
+1

latex interpreter fails

Aug 7, 2019
edited by Rom  Aug 7, 2019
edited by Rom  Aug 7, 2019
#2
+25648
+3

Find all solutions to the inequality $$\large{\dfrac{(2x-7)(x-3)}{x} \ge 0}$$ in interval notation

$$\begin{array}{|rcll|} \hline \mathbf{\dfrac{(2x-7)(x-3)}{x}} &\ge& \mathbf{0} \quad | \quad \boxed{x\neq 0 !} \\\\ \dfrac{(2x-7)(x-3)}{x} &\ge& 0 \quad | \quad \times x^2 \\ \dfrac{(2x-7)(x-3)x^2}{x} &\ge& 0\times x^2 \\ (2x-7)(x-3)x &\ge& 0 \\ 2\left(x-\dfrac{7}{2}\right)(x-3)x &\ge& 0 \quad | \quad : 2 \\ \left(x-\dfrac{7}{2}\right)(x-3)x &\ge& 0 \\ \mathbf{ \left(x-\dfrac{7}{2}\right)(x-3)(x-0) } &\ge& \mathbf{0} \\ \hline \end{array}$$

We create a sign table:

$$\begin{array}{|l|c|c|c|c|c|c|c|} \hline \text{Interval or position} : & (-\infty,0) & 0 & (0,3) & 3 & \left(3,\dfrac{7}{2}\right) &\dfrac{7}{2} & \left(\dfrac{7}{2},\infty\right) \\ \hline \text{sign of } (x-0): & - & 0 & + & + & + & + & + \\ \hline \text{sign of } (x-3): & - & - & - & 0 & + & + & + \\ \hline \text{sign of } \left(x-\dfrac{7}{2}\right): & - & - & - & - & - & 0 & + \\ \hline \text{sign of }\left(x-\dfrac{7}{2}\right)(x-3)(x-0): & - & \color{red}0 & \color{red}+ & \color{red}0 & - & \color{red}0 & \color{red}+ \\ \hline \end{array}$$

We can read the result:

in the interval $$[0,3]$$and $$\Big[\dfrac{7}{2},\infty\Big)$$ the left side of the inequality is positive or zero, and thus the inequality is true there.

But $$\boxed{x\neq 0 !}$$ and the solution is: $$\Big(0,3\Big]$$ and $$\Big[\dfrac{7}{2},\infty\Big)$$

Aug 7, 2019
edited by heureka  Aug 7, 2019
edited by heureka  Aug 7, 2019
edited by heureka  Aug 7, 2019
edited by heureka  Aug 7, 2019
edited by heureka  Aug 7, 2019
edited by heureka  Aug 7, 2019
edited by heureka  Aug 7, 2019
edited by heureka  Aug 7, 2019
#3
+112010
+3

I did not realize it was going to be so complicated.

Nicely done Heureka!

Here is the graph from Desmos graphing calculator

https://www.desmos.com/calculator/lcsrdedwxr

Aug 7, 2019
edited by Melody  Aug 7, 2019
#4
+25648
+1

Thank you, Melody !

heureka  Aug 7, 2019