To determine the tire size [tex]\( x \)[/tex] that provides the best performance for Michael's car, we start with the given equation:
[tex]\[
0.2(x - 25.5) + 0.3 = -0.2(x - 16)
\][/tex]
We'll solve this equation step-by-step.
### Step 1: Distribute the constants on both sides of the equation
On the left side:
[tex]\[
0.2(x - 25.5) + 0.3 = 0.2x - 0.2 \cdot 25.5 + 0.3 = 0.2x - 5.1 + 0.3
\][/tex]
Simplify the expression:
[tex]\[
0.2x - 5.1 + 0.3 = 0.2x - 4.8
\][/tex]
On the right side:
[tex]\[
-0.2(x - 16) = -0.2x + 0.2 \cdot 16 = -0.2x + 3.2
\][/tex]
So, we now have:
[tex]\[
0.2x - 4.8 = -0.2x + 3.2
\][/tex]
### Step 2: Combine like terms
To isolate [tex]\( x \)[/tex], we add [tex]\( 0.2x \)[/tex] to both sides of the equation:
[tex]\[
0.2x + 0.2x - 4.8 = -0.2x + 0.2x + 3.2
\][/tex]
This simplifies to:
[tex]\[
0.4x - 4.8 = 3.2
\][/tex]
### Step 3: Isolate the variable [tex]\( x \)[/tex]
Next, add 4.8 to both sides of the equation to move the constant term:
[tex]\[
0.4x - 4.8 + 4.8 = 3.2 + 4.8
\][/tex]
This simplifies to:
[tex]\[
0.4x = 8.0
\][/tex]
### Step 4: Solve for [tex]\( x \)[/tex]
Finally, divide both sides by 0.4 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{8.0}{0.4}
\][/tex]
[tex]\[
x = 20
\][/tex]
Therefore, the tire size that will yield the best performance for Michael's car is:
[tex]\[
20 \text{ inch tire}
\][/tex]
