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]
