Chapter 4.6, Problem 1.6P

Solution

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

  $x>-2.959$

Given:

An inequality $-284{\left(\frac{9}{7}\right)}^x>-135$.

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 $y=-284{\left(\frac{9}{7}\right)}^x$ . Now find the value of $y=-284{\left(\frac{9}{7}\right)}^x$ at different values of $x$.

$x$	$y$
$-2.5$	$-151.516$
$-2.6$	$-147.755$
$-2.7$	$-144.088$
$-2.8$	$-140.512$
$-2.9$	$-137.025$
$-3.0$	$-133.624$

According to the table, at $x=-2.9$ , the value of the function is less than $-135$ and at $x=-3.0$ the value of the function is greater than $-135$ . So, it can be concluded that the solution of inequality lies between $-2.9$ and $-3.0$.

Using a graph:

It is understood that $y=-284{\left(\frac{9}{7}\right)}^x$ and $y=-135$ . Substitute both functions in a graphing calculator to obtain the graph.

  

Both the functions intersect at the point $\left(-2.959,-135\right)$ . It means this point is common in both the functions. So, the solution of the inequality is $x>-2.959$.

Conclusion:

The solution of the inequality is $x>-2.959$.