Question

two inequalities are given. which number line shows the solution set for each inequality? drag and drop the correct response into each box in the table below.

Explanation:

Step1: Solve the inequality \(-x - 4\leq - 7\)

Add \(4\) to both sides: \(-x-4 + 4\leq-7 + 4\), which simplifies to \(-x\leq - 3\). Multiply both sides by \(-1\) (and reverse the inequality sign): \(x\geq3\). A closed - circle is used at \(x = 3\) (because the inequality is \(\geq\)) and the line extends to the right.

Step2: Solve the inequality \(4x+1\lt - 7\)

Subtract \(1\) from both sides: \(4x+1 - 1\lt-7 - 1\), so \(4x\lt-8\). Divide both sides by \(4\): \(x\lt - 2\). An open - circle is used at \(x=-2\) (because the inequality is \(\lt\)) and the line extends to the left.

Answer:

For \(-x - 4\leq - 7\), the solution set is \(x\geq3\) (the number line with a closed - circle at \(3\) and line to the right). For \(4x + 1\lt-7\), the solution set is \(x\lt - 2\) (the number line with an open - circle at \(-2\) and line to the left).