Find a number, delta, such that if |x-2|<delta, then |4x-8|<0.1

I don't have a clue how to begin this. My teacher just started off the lesson with this without any explanation so I am at a loss.

A step-by-step explanation would be very much appreciated!!

|4x-8| = 4|x-2|

So, if |4x-8| < 0.1
4|x-2| < 0.1
|x-2| < 0.025

So, x is less than 0.025 away from 2. That is,

1.975 < x < 2.025

Or, using the definition of |n|, either
(x-2) < 0.025 if x-2 >= 0
or
-(x-2) < 0.025 if x-2 < 0

So, if x-2 >= 0, x >= 2 and
x < 2.025

If x-2 < 0, x < 2, and
x-2 > -0.025
x > 1.975

So, 1.975 < x < 2.025

Actually, all that stuff above is irrelevant. The question was to find δ.

|4x-8| < .1
4|x-2| < 0.1
|x-2| < 0.025
so, δ = 0.025

To find the value of delta in the given inequality, we need to use the definition of a limit.

Step 1: Start with the inequality: |x - 2| < delta, and note that it represents the distance between x and 2.
Step 2: Simplify the inequality: |x - 2| < delta implies -delta < x - 2 < delta.
Step 3: Use this inequality to derive an expression for 4x - 8: multiply the inequality by 4, which gives -4delta < 4x - 8 < 4delta.
Step 4: To determine the required delta, we want to find a value such that |4x - 8| < 0.1. We can set up an inequality based on this requirement: |4x - 8| < 0.1 can be expressed as -0.1 < 4x - 8 < 0.1.
Step 5: Notice that these two inequalities, -4delta < 4x - 8 < 4delta and -0.1 < 4x - 8 < 0.1, are compatible with each other. This means that for any value of delta that satisfies -delta < x - 2 < delta, it will also satisfy -0.1 < 4x - 8 < 0.1.
Step 6: So, to find the value of delta, we need to solve the inequality -4delta < 4x - 8 < 4delta simultaneously with -0.1 < 4x - 8 < 0.1.
Step 7: Subtract 4x - 8 from each inequality of -4delta < 4x - 8 < 4delta to obtain -4delta - (4x - 8) < 0 < 4delta - (4x - 8). Simplifying, we get -4delta - 4x + 8 < 0 < 4delta - 4x + 8.
Step 8: Next, let's break these two inequalities into separate ones. We have -4delta - 4x + 8 < 0 and 0 < 4delta - 4x + 8.
Step 9: For the first inequality, -4delta - 4x + 8 < 0, add 4delta - 8 to both sides to get -4x < 4delta, then divide by -4, which flips the inequality sign to get x > -delta.
Step 10: For the second inequality, 0 < 4delta - 4x + 8, add 4x - 8 to both sides, which gives 4delta > 4x, and then divide by 4, resulting in delta > x.
Step 11: Combining the results from step 9 and step 10, we have x > -delta and delta > x, which means that the value of delta must be positive.
Step 12: Based on the previous step, we can conclude that delta can be any positive value. Therefore, there is no specific value of delta that satisfies the given inequality |4x - 8| < 0.1 for all x satisfying |x - 2| < delta.

In summary, according to the given inequality, there is no specific value of delta that satisfies the condition |4x - 8| < 0.1 for all x such that |x - 2| < delta.