**Graphing the solutions of an inequality :**

A solution of an inequality that contains a variable is any value of the variable that makes the inequality true.

For example, 7 is a solution of x > -2, since 7 > -2 is a true statement

**Example 1 :**

Graph the solutions of the inequality y ≤ -3. Check the solutions.

**Solution : **

**Step 1 :**

Draw a solid circle at -3 to show that -3 is a solution.

**Step 2 :**

Shade the number line to the left of -3 to show that numbers less than -3 are solutions.

(Use a solid circle for an inequality that uses ≥ or ≤)

**Step 3 :**

Check your solution.

Choose a number that is on the shaded section of the number line, such as -4.

Substitute -4 for y.

-4 ≤ -3

-4 is less than -3, so -4 is a solution.

**Step 4 :**

Let us prove that -3 is a solution of the inequality y ≤ -3.

In the given inequality, plug y = -3.

Then, we have

-3 ≤ -3 ---> (-3 is less than or equal to -3) ?

Is the answer for the above question is "yes or "no" ?

The answer for the above question is "Yes".

Because, -3 is equal to -3.

Hence, -3 is a solution to the inequality y ≤ -3.

**Example 2 :**

Graph the solutions of the inequality 1 < m. Check the solutions.

**Solution : **

**Step 1 :**

Draw an empty circle at 1 to show that 1 is not a solution.

**Step 2 :**

Shade the number line to the right of 1 to show that numbers greater than 1 are solutions.

(Use an open circle for an inequality that uses > or <)

**Step 3 :**

Check your solution.

Choose a number that is on the shaded section of the number line, such as 2.

Substitute -4 for y.

1 < 2

1 is less than 2, so 2 is a solution.

**Step 4 :**

Let us prove that 1 is not a solution of the inequality 1 < m.

In the given inequality, plug m = 1.

Then, we have

1 < 1 ---> (1 is less than 1) ?

Is the answer for the above question is "yes or "no" ?

The answer for the above question is "No".

Because, 1 is equal to 1.

Hence, 1 is not a solution to the inequality 1 < m.

**Example 3 :**

Graph the solutions of the inequality t ≤ -4. Check the solutions.

**Solution : **

**Step 1 :**

Draw a solid circle at -4 to show that -4 is a solution.

**Step 2 :**

Shade the number line to the left of -4 to show that numbers less than -4 are solutions.

(Use a solid circle for an inequality that uses ≥ or ≤)

**Step 3 :**

Check your solution.

Choose a number that is on the shaded section of the number line, such as -5.

Substitute -5 for t.

-5 ≤ -4

-5 is less than -4, so -5 is a solution.

**Step 4 :**

Let us prove that -4 is a solution of the inequality t ≤ -4.

In the given inequality, plug t = -4.

Then, we have

-4 ≤ -4 ---> (-4 is less than or equal to -4) ?

Is the answer for the above question is "yes or "no" ?

The answer for the above question is "Yes".

Because, -4 is equal to -4.

Hence, -4 is a solution to the inequality t ≤ -4.

After having gone through the stuff given above, we hope that the students would have understood "Graphing the solutions of an inequality".

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