Simplify 0.8x + (1 - x)(0.2).

Step-by-step solution

1. 1

   Recognise the structure before simplifying. This is a weighted average: a fraction $x$ of something contributes at rate $0.8$, and the remaining fraction $1-x$ contributes at rate $0.2$. Expressions of this shape appear in mixture, probability, and blended-rate problems, and knowing that tells you the simplified form must be linear in $x$.

2. 2

   Distribute the 0.2 across the parentheses.

   $0.2\cdot1=0.2,\qquad 0.2\cdot(-x)=-0.2x$

   so the expression becomes

   $0.8x+0.2-0.2x$

3. 3

   Group the like terms. The two $x$ terms combine; the lone constant does not:

   $(0.8x-0.2x)+0.2$

4. 4

   Combine and state the result.

   $0.8x-0.2x=0.6x\qquad\Longrightarrow\qquad 0.6x+0.2$

   The slope $0.6$ is exactly the gap between the two rates, $0.8-0.2$, and the intercept $0.2$ is the value when $x=0$.

5. 5

   Sanity-check the endpoints. At $x=0$ the expression should be the pure low rate: $0.6(0)+0.2=0.2$ ✓. At $x=1$ it should be the pure high rate: $0.6(1)+0.2=0.8$ ✓. Both endpoints land exactly where the weighted-average reading predicts.

6. 6

   Verify across random values. Comparing the original and simplified forms at $40$ random values of $x$ drawn from $[-5,5]$ gives agreement to machine precision at every point ✓, confirming the result holds outside $[0,1]$ too.