We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
152
4
avatar

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

 Aug 7, 2019
 #1
avatar+6045 
+1

latex interpreter fails

 Aug 7, 2019
edited by Rom  Aug 7, 2019
edited by Rom  Aug 7, 2019
 #2
avatar+23342 
+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 \(\Big[\dfrac{7}{2},\infty\Big)\)

 

 

laugh

 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
avatar+105689 
+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
avatar+23342 
+1

Thank you, Melody !

 

laugh

heureka  Aug 7, 2019

9 Online Users

avatar