Consider the equation below:

[tex]\[ 3^{-x} - 3 = 4^x - 1 \][/tex]

Select the approximate solution to the equation from the table below:

\begin{tabular}{|c|c|c|}
\hline
[tex]$x$[/tex] & [tex]$3^{-x} - 3$[/tex] & [tex]$4^x - 1$[/tex] \\
\hline
-1.75 & 3.83 & -0.91 \\
\hline
-1.5 & 2.19 & -0.88 \\
\hline
-1.25 & 0.95 & -0.82 \\
\hline
-1 & 0 & -0.75 \\
\hline
-0.75 & -0.72 & -0.65 \\
\hline
-0.5 & -1.27 & -0.50 \\
\hline
-0.25 & -1.68 & -0.29 \\
\hline
\end{tabular}



Answer :

To find the approximate solution to the equation [tex]\(3^{(-x)} - 3 = 4^x - 1\)[/tex] using the given table, let's examine each value of [tex]\(x\)[/tex] step-by-step:

We are tasked with finding the value of [tex]\(x\)[/tex] that makes the left-hand side ([tex]\(3^{(-x)} - 3\)[/tex]) and the right-hand side ([tex]\(4^x - 1\)[/tex]) as close to each other as possible.

Given the table:
[tex]\[ \begin{tabular}{|c|c|c|} \hline $x$ & $3^{(-x)} - 3$ & $4^x - 1$ \\ \hline -1.75 & 3.83 & -0.91 \\ \hline -1.5 & 2.19 & -0.88 \\ \hline -1.25 & 0.95 & -0.82 \\ \hline -1 & 0 & -0.75 \\ \hline -0.75 & -0.72 & -0.65 \\ \hline -0.5 & -1.27 & -0.50 \\ \hline -0.25 & -1.68 & -0.29 \\ \hline \end{tabular} \][/tex]

Let’s compute the absolute differences between [tex]\(3^{(-x)} - 3\)[/tex] and [tex]\(4^x - 1\)[/tex] for each provided [tex]\(x\)[/tex]:

1. For [tex]\(x = -1.75\)[/tex]:
[tex]\[ \left|3.83 - (-0.91)\right| = 3.83 + 0.91 = 4.74 \][/tex]

2. For [tex]\(x = -1.5\)[/tex]:
[tex]\[ \left|2.19 - (-0.88)\right| = 2.19 + 0.88 = 3.07 \][/tex]

3. For [tex]\(x = -1.25\)[/tex]:
[tex]\[ \left|0.95 - (-0.82)\right| = 0.95 + 0.82 = 1.77 \][/tex]

4. For [tex]\(x = -1\)[/tex]:
[tex]\[ \left|0 - (-0.75)\right| = 0 + 0.75 = 0.75 \][/tex]

5. For [tex]\(x = -0.75\)[/tex]:
[tex]\[ \left|-0.72 - (-0.65)\right| = \left|-0.72 + 0.65\right| = \left|-0.07\right| = 0.07 \][/tex]

6. For [tex]\(x = -0.5\)[/tex]:
[tex]\[ \left|-1.27 - (-0.50)\right| = \left|-1.27 + 0.50\right| = \left|-0.77\right| = 0.77 \][/tex]

7. For [tex]\(x = -0.25\)[/tex]:
[tex]\[ \left|-1.68 - (-0.29)\right| = \left|-1.68 + 0.29\right| = \left|-1.39\right| = 1.39 \][/tex]

By comparing the absolute differences, we find that the smallest difference is [tex]\(0.07\)[/tex] when [tex]\(x = -0.75\)[/tex]. Therefore, the approximate solution to the equation [tex]\(3^{(-x)} - 3 = 4^x - 1\)[/tex] is:
[tex]\[ x = -0.75 \][/tex]

Thus, the correct solution in the table is:
[tex]\[ -0.75 \][/tex]