Question

use successive approximations to estimate a solution of this equation:
$-2x + 10 = 0.3^x$
first, complete the table. round your answers to the nearest tenth.

$x$	$-2x + 10$	$0.3^x$
$-3$	$16$	$37.0$
$-2$	$14$	$11.1$
$-2.3$	$14.6$	$15.9$
$-2.2$	$14.4$	$14.1$
$-2.25$	$square$	$square$

now, use the table to estimate one solution, rounded to the nearest tenth.
$x \approx \square$

Explanation:

Step1: Calculate $-2x+10$ for $x=-2.25$

Substitute $x=-2.25$ into the linear expression:
$$-2(-2.25) + 10 = 4.5 + 10 = 14.5$$

Step2: Calculate $0.3^x$ for $x=-2.25$

Rewrite the expression using negative exponent rules, then compute:
$$0.3^{-2.25} = \frac{1}{0.3^{2.25}} \approx \frac{1}{0.067} \approx 14.9$$

Step3: Estimate the solution

Find where $-2x+10$ and $0.3^x$ are closest (cross). At $x=-2.2$, $-2x+10=14.4$, $0.3^x=14.1$; at $x=-2.3$, $-2x+10=14.6$, $0.3^x=15.9$. The values are closest at $x=-2.2$.

Answer:

Completed table values for $x=-2.25$:
$-2x+10 = 14.5$, $0.3^x \approx 14.9$
Estimated solution: $x \approx -2.2$