We are given that $y = 5x + 4$ and it is required that $y$ be within 0.007 units of 8. This can be mathematically expressed as:

$|y - 8| \leq 0.007$

Substituting $y = 5x + 4$ into the inequality:

$|5x + 4 - 8| \leq 0.007$

Simplify the expression inside the absolute value:

$|5x - 4| \leq 0.007$

To solve this inequality, we remove the absolute value by setting up the two cases:

1. $5x - 4 \leq 0.007$
2. $5x - 4 \geq -0.007$

Case 1: $5x - 4 \leq 0.007$

Add 4 to both sides:

$5x \leq 4.007$

Divide by 5:

$x \leq \frac{4.007}{5} = 0.8014$

Case 2: $5x - 4 \geq -0.007$

Add 4 to both sides:

$5x \geq 3.993$

Divide by 5:

$x \geq \frac{3.993}{5} = 0.7986$

Final Solution:

Combining the two inequalities, the solution for $x$ is:

$0.7986 \leq x \leq 0.8014$

This is the range of values for $x$ where $y$ will be within 0.007 units of 8.

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. What would be the solution if $y = 5x + 4$ needs to be within 0.01 units of 10?
2. How do you solve inequalities that do not involve absolute values?
3. Can you graph the solution set on a number line?
4. What happens if the required range was exactly 0.007 units from 8?
5. How does changing the coefficient of $x$ in $y = 5x + 4$ affect the solution?

Tip: When dealing with inequalities involving absolute values, remember to set up two cases—one for the positive and one for the negative scenarios—before solving.