Suppose that y= 5 x plus 4 and it is required that y be within 0.007 units of 8. For what values of x will this be​ true?

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.

Learn how to solve linear equations involving absolute value inequalities with a detailed step-by-step solution. This problem covers key algebraic concepts such as linear equations, inequalities, and absolute value. Suitable for students in Grades 7-9.

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