Solve the inequality and express the answer in interval notation:
$2|x-4|-7 \leq 1$
Interval Notation:
The objective is to solve the inequality \(2|x - 4| - 7 \leq 1\) for \(x\). This inequality involves an absolute value term, which represents the distance of the expression \((x - 4)\) from zero on a number line. To isolate \(x\), we first need to isolate the absolute value term.
To isolate the absolute value term, we start by adding \(7\) to both sides of the inequality:
\[2|x - 4| \leq 1 + 7\]
\[2|x - 4| \leq 8\]
Next, divide both sides by \(2\) to obtain:
\[|x - 4| \leq 4\]
The inequality \(|x - 4| \leq 4\) states that the distance between \(x\) and \(4\) is at most \(4\). By the definition of absolute value inequalities, \(|u| \leq a\) is equivalent to the compound inequality \(-a \leq u \leq a\). Applying this to our expression:
\[-4 \leq x - 4 \leq 4\]
To solve for \(x\), add \(4\) to all parts of the compound inequality:
\[-4 + 4 \leq x - 4 + 4 \leq 4 + 4\]
\[0 \leq x \leq 8\]
This gives us the set of all real numbers \(x\) such that \(x\) is between \(0\) and \(8\), inclusive.
The solution \(0 \leq x \leq 8\) describes a closed interval on the number line. In interval notation, we use square brackets to indicate that the endpoints \(0\) and \(8\) are included in the solution set. Therefore, the solution is \([0, 8]\).
Consider a manufacturing process where a part needs to have a length of 4 cm, with a tolerance of up to 4 cm deviation from this target. The inequality \(|x - 4| \leq 4\) represents the range of acceptable lengths, where \(x\) is the actual length of the part. By solving this, we find that any length between 0 cm and 8 cm is within the allowed tolerance, ensuring quality control in the manufacturing process.