**Solving inequalities using multiplication and division :**

We can use properties of inequality to solve inequalities involving multiplication and division with rational numbers.

**Example 1 : **

Solve the inequality given below. Graph and check the solution.

y/3 ≥ 5

**Solution :**

**Step 1 : **

Solve the inequality

y/3 ≥ 5

Use the Multiplication Property of Inequality.

Multiply both sides by 3.

3.(y/3) ≥ (5).3

y ≥ 15

**Step 2 : **

Graph the solution.

**Step 3 :**

Check the solution by substituting a solution from the shaded part of the graph into the original inequality. For convenience, choose a multiple of 3.

Substitute 18 for x into y/3 ≥ 5

18/3 ≥ 5

6 ≥ 5

The inequality is true.

**Example 2 : **

Solve the inequality given below. Graph and check the solution.

-4x > 52

**Solution :**

**Step 1 : **

Solve the inequality

-4x > 52

Use the Division Property of Inequality.

Divide both sides by -4.

(-4x) / (-4) < 52 / (-4)

x < -13

**Step 2 : **

Graph the solution.

**Step 3 :**

Check the solution. Substitute a solution from the shaded part of your number line into the original inequality.

Substitute -15 for x into -4x > 52.

-4.(-15) > 52

60 > 52

The inequality is true.

**Example 3 : **

Solve the inequality given below. Graph and check the solution.

-10y < 60

**Solution :**

**Step 1 : **

Solve the inequality

-10y < 60

Use the Division Property of Inequality.

Divide both sides by -10.

(-10y) / (-10) > 60 / (-10)

y > -6

**Step 2 : **

Graph the solution.

**Step 3 :**

Check the solution. Substitute a solution from the shaded part of your number line into the original inequality.

Substitute -2 for y into -10y < 60.

(-10).(-2) < 60

20 < 60

The inequality is true.

**Example 4 : **

Solve the inequality given below. Graph and check the solution.

7 ≥ -t/6

**Solution :**

**Step 1 : **

Solve the inequality

7 ≥ -t/6

Use the Multiplication Property of Inequality.

Multiply both sides by -6.

7.(-6) ≤ (-t/6).(-6)

-42 ≤ t

or t ≥ -42

**Step 2 : **

Graph the solution.

**Step 3 :**

Check the solution. Substitute a solution from the shaded part of your number line into the original inequality.

Substitute -40 for t into 7 ≥ -t/6.

7 ≥ -(-40)/6

7 ≥ 20/3

7 ≥ 6.67

The inequality is true.

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