Question

one-step inequalities
solve each inequality and graph its solution.

1. $-12 < x - 7$
2. $-1 + r \geq 4$
3. $n - 6 \leq -14$
4. $b - 7 < -12$
5. $a - 17 > -16$
6. $15 + x \leq 0$
7. $3 + v \leq -9$
8. $8 \geq n - 6$
9. $-3x \geq 3$
10. $\frac{n}{3} > 3$
11. $\frac{k}{4} < -4$
12. $-9x \geq -90$

Answer:

Let's solve each one - step inequality:

1. Solve \(- 12\lt x - 7\)

Step 1: Isolate the variable \(x\)

Add 7 to both sides of the inequality. According to the addition property of inequalities, if \(a\lt b\), then \(a + c\lt b + c\) for any real number \(c\).
\(-12 + 7\lt x-7 + 7\)

Step 2: Simplify the inequality

Simplify both sides: \(-5\lt x\) or \(x\gt - 5\)

2. Solve \(1 + r\geq4\)

Step 1: Isolate the variable \(r\)

Subtract 1 from both sides of the inequality. If \(a + b\geq c\), then \(a\geq c - b\) (subtraction property of inequalities).
\(1+r - 1\geq4 - 1\)

Step 2: Simplify the inequality

Simplify both sides: \(r\geq3\)

3. Solve \(n - 6\leq-14\)

Step 1: Isolate the variable \(n\)

Add 6 to both sides of the inequality. Using the addition property of inequalities (\(a - b\leq c\Rightarrow a\leq c + b\))
\(n-6 + 6\leq-14 + 6\)

Step 2: Simplify the inequality

Simplify both sides: \(n\leq - 8\)

4. Solve \(b - 7\lt-12\)

Step 1: Isolate the variable \(b\)

Add 7 to both sides of the inequality. By the addition property of inequalities (\(a - b\lt c\Rightarrow a\lt c + b\))
\(b-7 + 7\lt-12 + 7\)

Step 2: Simplify the inequality

Simplify both sides: \(b\lt - 5\)

5. Solve \(a - 17\gt-16\)

Step 1: Isolate the variable \(a\)

Add 17 to both sides of the inequality. Using the addition property of inequalities (\(a - b\gt c\Rightarrow a\gt c + b\))
\(a-17 + 17\gt-16 + 17\)

Step 2: Simplify the inequality

Simplify both sides: \(a\gt1\)

6. Solve \(15 + x\leq0\)

Step 1: Isolate the variable \(x\)

Subtract 15 from both sides of the inequality. By the subtraction property of inequalities (\(a + b\leq c\Rightarrow a\leq c - b\))
\(15+x - 15\leq0 - 15\)

Step 2: Simplify the inequality

Simplify both sides: \(x\leq - 15\)

7. Solve \(3 + v\leq-9\)

Step 1: Isolate the variable \(v\)

Subtract 3 from both sides of the inequality. Using the subtraction property of inequalities (\(a + b\leq c\Rightarrow a\leq c - b\))
\(3 + v-3\leq-9 - 3\)

Step 2: Simplify the inequality

Simplify both sides: \(v\leq - 12\)

8. Solve \(8\geq n - 6\)

Step 1: Isolate the variable \(n\)

Add 6 to both sides of the inequality. By the addition property of inequalities (\(a\geq b - c\Rightarrow a + c\geq b\))
\(8 + 6\geq n-6 + 6\)

Step 2: Simplify the inequality

Simplify both sides: \(14\geq n\) or \(n\leq14\)

9. Solve \(-3x\geq3\)

Step 1: Isolate the variable \(x\)

Divide both sides of the inequality by - 3. When dividing an inequality by a negative number, the direction of the inequality sign changes. If \(a\gt0\) and \(-ax\geq b\), then \(x\leq-\frac{b}{a}\)
\(\frac{-3x}{-3}\leq\frac{3}{-3}\)

Step 2: Simplify the inequality

Simplify both sides: \(x\leq - 1\)

10. Solve \(\frac{n}{3}\gt3\)

Step 1: Isolate the variable \(n\)

Multiply both sides of the inequality by 3. According to the multiplication property of inequalities, if \(a\gt0\) and \(\frac{x}{a}\gt b\), then \(x\gt ab\)
\(\frac{n}{3}\times3\gt3\times3\)

Step 2: Simplify the inequality

Simplify both sides: \(n\gt9\)

11. Solve \(\frac{k}{4}\lt - 4\)

Step 1: Isolate the variable \(k\)

Multiply both sides of the inequality by 4. Using the multiplication property of inequalities, if \(a\gt0\) and \(\frac{x}{a}\lt b\), then \(x\lt ab\)
\(\frac{k}{4}\times4\lt - 4\times4\)

Step 2: Simplify the inequality

Simplify both sides: \(k\lt - 16\)

12. Solve \(-9x\geq - 90\)

Step 1: Isolate the variable \(x\)

Divide both sides of the inequality by - 9. When dividing an inequality by a negative number, the direction of the inequality sign changes. If \(a\gt0\) and \(-ax\geq b\), then \(x\leq-\frac{b}{a}\)
\(\frac{-9x}{-9}\leq\frac{-90}{-9}\)

Step 2: Simplify the inequality

Simplify both sides: \(x\leq10\)

Graphing the solutions:

• For \(x\gt - 5\): Draw an open circle at \(x=-5\) and a line with an arrow pointing to the right.
• For \(r\geq3\): Draw a closed circle at \(r = 3\) and a line with an arrow pointing to the right.
• For \(n\leq - 8\): Draw a closed circle at \(n=-8\) and a line with an arrow pointing to the left.
• For \(b\lt - 5\): Draw an open circle at \(b = - 5\) and a line with an arrow pointing to the left.
• For \(a\gt1\): Draw an open circle at \(a = 1\) and a line with an arrow pointing to the right.
• For \(x\leq - 15\): Draw a closed circle at \(x=-15\) and a line with an arrow pointing to the left.
• For \(v\leq - 12\): Draw a closed circle at \(v=-12\) and a line with an arrow pointing to the left.
• For \(n\leq14\): Draw a closed circle at \(n = 14\) and a line with an arrow pointing to the left.
• For \(x\leq - 1\): Draw a closed circle at \(x=-1\) and a line with an arrow pointing to the left.
• For \(n\gt9\): Draw an open circle at \(n = 9\) and a line with an arrow pointing to the right.
• For \(k\lt - 16\): Draw an open circle at \(k=-16\) and a line with an arrow pointing to the left.
• For \(x\leq10\): Draw a closed circle at \(x = 10\) and a line with an arrow pointing to the left.