Chapter 4.6, Problem 1.9P

Solution

To solve: The given logarithmic inequality using table and graph.

  $0<x<36$

Given:

An inequality ${\log }_{6}x+9\le 11$.

Concept used:

1. To obtain obtain the solution using a table, look for the value of $x$ for which the function produces the value closer to the numerical part of inequality.
2. The $x$-coordinate of the intersection point of both functions represents the solution of the equation.

Calculation:

Using a table:

Consider that both sides of the inequality represent a function. So, it can be assumed that $y={\log }_{6}x+9$ . Now find the value of $y={\log }_{6}x+9$ at different values of $x$.

$x$	$y$
$30$	$10.898$
$31$	$10.917$
$32$	$10.934$
$33$	$10.951$
$34$	$10.968$
$35$	$10.984$
$36$	$11$

According to the table, at $x=36$ , the value of the function is equal to $11$ and according to inequality, the value must be either less than or equal to $11$ so the solution of the inequality is $0<x<36$.

Using a graph:

It is understood that $y={\log }_{6}x+9$ and $y=11$ . Substitute both functions in a graphing calculator to obtain the graph.

  

Both the functions intersect at the point $\left({\log }_{6}x+9,11\right)$ . It means this point is common in both the functions. So, the solution of the inequality is $0<x<36$.

Conclusion:

The solution of the inequality is $0<x<36$.