Absolutely, I'd be happy to break down the steps for this inequality.
1) We start with the inequality |2x - 6| ≤ 4.
An absolute value inequality of the form |a| ≤ b means that a is in the closed interval from -b to b. This implies that -b ≤ a ≤ b.
So, we rewrite the given inequality as -4 ≤ 2x - 6 ≤ 4.
2) Now we start solving the inequality. Let's divide the inequality into two parts: one for the left and one for the right and solve them separately.
For the left part: -4 ≤ 2x - 6.
By adding 6 to both sides, the inequality becomes 2 ≤ 2x
When we divide by 2, it simplifies to 1 ≤ x.
For the right side: 2x - 6 ≤ 4.
Add 6 to both sides to get 2x ≤ 10.
After dividing by 2, it simplifies to x ≤ 5.
3) Combine both solutions together:
Therefore the final solution for the inequality |2x - 6| ≤ 4 is 1 ≤ x ≤ 5.
This means that any values for x within this interval, including 1 and 5 themselves, are solutions of the inequality.
In other words, all the numbers between 1 and 5 (inclusive) satisfy the given inequality.