Chapter 4.6, Problem 1.4P

Solution

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

  $x\le 20.756$

Given:

An inequality $-63{\left(0.96\right)}^x<-27$.

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 $y=-63{\left(0.96\right)}^x$ . Now find the value of $y=-63{\left(0.96\right)}^x$ at different values of $x$.

$x$	$y$
$10$	$-41.884$
$15$	$-34.151$
$20$	$-27.846$
$25$	$-22.705$

According to the table, at $x=20$ , the value of the function is closer to the $-27$ . The values which are less than $20$ satisfy the inequality condition and the values which are greater than $20$ does not satisfy the inequality condition. So, the solution of the inequality is $x\le 20$

Using a graph:

It is understood that $y=-63{\left(0.96\right)}^x$ and $y=-27$ . Substitute both functions in a graphing calculator to obtain the graph.

  

Both the functions intersect at the point $\left(20.756,-27\right)$ . It means this point is common in both the functions. So, the solution of the inequality is $x\le 20.756$.

Conclusion:

The solution of the inequality is $x\le 20.756$.