Question

solving one - step inequalities
solve each inequality. then graph the solution set on the number line.
x - 6≥ - 14
3p≤ - 21
y - 3.25>6.75
\\(\frac{k}{-4}>-5\\)

• 8j<16

z + \\(\frac{7}{4}\\)≥ - \\(\frac{11}{4}\\)

Explanation:

Step1: Solve \(x - 6\geq - 14\)

Add 6 to both sides: \(x-6 + 6\geq - 14+6\), so \(x\geq - 8\). On the number - line, we use a closed circle at - 8 (because of \(\geq\)) and draw an arrow to the right.

Step2: Solve \(\frac{k}{-4}>-5\)

Multiply both sides by - 4. Since we are multiplying by a negative number, we flip the inequality sign: \(k < (-5)\times(-4)\), so \(k < 20\). On the number - line, we use an open circle at 20 (because of \(<\)) and draw an arrow to the left.

Step3: Solve \(3p\leq - 21\)

Divide both sides by 3: \(p\leq\frac{-21}{3}\), so \(p\leq - 7\). On the number - line, we use a closed circle at - 7 (because of \(\leq\)) and draw an arrow to the left.

Step4: Solve \(-8j < 16\)

Divide both sides by - 8. Since we are dividing by a negative number, we flip the inequality sign: \(j>-2\). On the number - line, we use an open circle at - 2 (because of \(>\)) and draw an arrow to the right.

Step5: Solve \(y - 3.25>6.75\)

Add 3.25 to both sides: \(y-3.25 + 3.25>6.75 + 3.25\), so \(y > 10\). On the number - line, we use an open circle at 10 (because of \(>\)) and draw an arrow to the right.

Step6: Solve \(z+\frac{7}{4}\geq-\frac{1}{4}\)

Subtract \(\frac{7}{4}\) from both sides: \(z+\frac{7}{4}-\frac{7}{4}\geq-\frac{1}{4}-\frac{7}{4}\), \(z\geq\frac{-1 - 7}{4}\), so \(z\geq - 2\). On the number - line, we use a closed circle at - 2 (because of \(\geq\)) and draw an arrow to the right.

Answer:

1. \(x\geq - 8\), number - line: closed circle at - 8, arrow to the right.
2. \(k < 20\), number - line: open circle at 20, arrow to the left.
3. \(p\leq - 7\), number - line: closed circle at - 7, arrow to the left.
4. \(j>-2\), number - line: open circle at - 2, arrow to the right.
5. \(y > 10\), number - line: open circle at 10, arrow to the right.
6. \(z\geq - 2\), number - line: closed circle at - 2, arrow to the right.