Question

solve and graph the solution set. -3 ≤ (x + 1)/-2 ≤ 5. graph the solution. choose the correct graph below. write the solution set using set - builder notation. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. {x| (use one inequality to express your answer if possible. for answers with more than one inequality that cannot be expressed by a single inequality, separate the inequalities by a comma. do not include the word or in your answer.)} b. the solution set is all real numbers. c. the solution set is the empty set. write the solution set using interval notation. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the solution is (type your answer in interval notation. simplify your answer. use integers or fractions for any numbers in the inequality.) b. the solution set is the empty set.

Explanation:

Step1: Solve the compound - inequality

We have \(-3\leq\frac{x + 1}{-2}\leq5\). First, multiply each part of the compound - inequality by \(-2\). Remember that when multiplying or dividing an inequality by a negative number, the direction of the inequality signs changes. So we get \(6\geq x + 1\geq - 10\).

Step2: Subtract 1 from each part

Subtract 1 from all parts of the inequality \(6-1\geq x+1 - 1\geq - 10 - 1\), which simplifies to \(5\geq x\geq - 11\) or \(-11\leq x\leq5\).

Step3: Write in set - builder and interval notation

In set - builder notation, the solution set is \(\{x|-11\leq x\leq5\}\). In interval notation, the solution set is \([-11,5]\).

Step4: Graph the solution

The graph of the solution \(-11\leq x\leq5\) is a number line with a closed - circle at \(-11\) and a closed - circle at 5, and the line segment between them is shaded.

Answer:

Set - builder notation: \(\{x|-11\leq x\leq5\}\)
Interval notation: \([-11,5]\)
Graph: A number line with closed - circles at \(-11\) and 5 and the line segment between them shaded.