Chapter 4.6, Problem 1.3P

Solution

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

  $x\le 1.51$

Given:

An inequality $244{\left(0.35\right)}^x\ge 50$.

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 $y=244{\left(0.35\right)}^x$ . Now find the value of $y=244{\left(0.35\right)}^x$ at different values of $x$.

$x$	$y$
$1.0$	$85.4$
$1.1$	$76.889$
$1.2$	$69.226$
$1.3$	$62.327$
$1.4$	$56.116$
$1.5$	$50.523$
$1.6$	$45.488$

According to the table, at $x=1.5$ , the value of the function is closer to the $50$ . The values which are less than $1.5$ satisfy the inequality condition and the values which are greater than $1.5$ does not satisfy the inequality condition. So, the solution of the inequality is $x\le 1.5$

Using a graph:

It is understood that $y=244{\left(0.35\right)}^x$ and $y=50$ . Substitute both functions in a graphing calculator to obtain the graph.

  

Both the functions intersect at the point $\left(1.51,50\right)$ . It means this point is common in both the functions. So, the solution of the inequality is $x\le 1.51$.

Conclusion:

The solution of the inequality is $x\le 1.51$.