Question

Solve each inequality algebraically.

$6x-5<\frac{6}{x}$

Short answer

Solution set and the interval is:
$$\begin{gathered} \left\{x\right|-\frac{2}{3}<x<\frac{3}{2}\left\} \\ \right(-\frac{2}{3},\frac{3}{2}) \end{gathered}$$

Step-by-step solution

Step 1. Change the inequality to equal to zero. 

Change the inequality equal to zero to make it easier and then get the value of x

First, arrange the inequality,

$$\begin{gathered} 6x-5<\frac{6}{x} \\ 6x-5-\frac{6}{x}<0 \\ 6x^2-5x-6<0 \end{gathered}$$ Make it equal to zero,

$$\begin{gathered} 6x^2-5x-6=0 \\ (3x+2)(2x-3)=0 \\ 3x+2=0;2x-3=0 \\ x=-\frac{2}{3};x=\frac{3}{2} \end{gathered}$$

Step 2. Form the intervals

From the obtained values of x, we can form the interval,

So the interval we have is:

$(-\infty ,-\frac{2}{3})\cup (-\frac{2}{3},\frac{3}{2})\cup (\frac{3}{2},\infty )$

Step 3. Form the table

Since, $f\left(x\right)<0$, so is negative.

check a number in each interval and evaluate the function to see whether it is satisfying the function.

Interval	Number chosen	Resultant
$(-\infty ,-\frac{2}{3})$	$-2$	$28<0$
$(-\frac{2}{3},\frac{3}{2})$	$0$	$-6<0$
$(\frac{3}{2},\infty )$	$2$	$8<0$

Step 4. Solution set and interval

Since,

$-6<0$satisfy $f$. Thus,

role="math" localid="1646201857095" $$\begin{gathered} \left\{x\right|-\frac{2}{3}<x<\frac{3}{2}\left\} \\ \right(-\frac{2}{3},\frac{3}{2}) \end{gathered}$$