Question

give the solution set in both interval and graph form.\\(|10 - x| \leq 1\\)\\(\quad\\)select the correct choice below and, if necessary, fill in the answer box to complete your choice.\\(\bigcirc\\) a. the solution set is \\(\square\\).\\(\quad\\) (type your answer in interval notation.)\\(\bigcirc\\) b. the solution set is \\(\varnothing\\).

Explanation:

Step1: Recall the absolute value inequality rule

For \(|a| \leq b\) (where \(b \geq 0\)), it is equivalent to \(-b \leq a \leq b\). Here, \(a = 10 - x\) and \(b = 1\), so we have \(-1 \leq 10 - x \leq 1\).

Step2: Solve the left - hand inequality

Start with \(-1 \leq 10 - x\). Subtract 10 from both sides: \(-1-10 \leq -x\), which simplifies to \(-11 \leq -x\). Multiply both sides by - 1 (and reverse the inequality sign): \(11\geq x\) or \(x\leq11\).

Step3: Solve the right - hand inequality

Now, take \(10 - x \leq 1\). Subtract 10 from both sides: \(-x \leq 1 - 10=-9\). Multiply both sides by - 1 (and reverse the inequality sign): \(x\geq9\).

Step4: Combine the two inequalities

From the two steps above, we have \(9\leq x\leq11\). In interval notation, this is \([9,11]\).

Answer:

A. The solution set is \([9, 11]\)