- Writing, Graphing and Solving Inequalities
- Home
- Translating a Sentence by Using an Inequality Symbol
- Translating a Sentence into a One-Step Inequality
- Introduction to Identifying Solutions to an Inequality
- Writing an Inequality for a Real-World Situation
- Graphing a Linear Inequality on the Number Line
- Writing an Inequality Given a Graph on the Number Line
- Identifying Solutions to a One-Step Linear Inequality
- Additive Property of Inequality with Whole Numbers
- Multiplicative Property of Inequality with Whole Numbers
- Solving a Two-Step Linear Inequality with Whole Numbers
- Solving a Word Problem Using a One-Step Linear Inequality

- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who

We represent inequalities by using a number line in this lesson.

In the examples below, we show the range of true values for a given inequality.

We use an

**open dot**to represent < and > relationships; this symbol indicates that the point on the number line is not included within the range of possible values for the inequality.We use a

**closed dot**to represent ≤ and ≥, when the two sides of the inequality could be equal.

**Number Line**

We recall that a number line is a horizontal line that has points, which correspond to numbers. The points are spaced according to the value of the number they correspond to; the points are equally spaced in a number line containing only whole numbers or integers.

**Graph of the point “3”**

We graph numbers by representing them as points on the number line. For example, we graph "3" on the number line as shown below −

**Graph of the Inequality x ≤ 3**

We can also graph inequalities on the number line. The following graph represents the inequality x ≤ 3. The dark line represents all the numbers that satisfy x ≤ 3. If we pick any number on the dark line and plug it in for x, the inequality will be true.

**Graph of the Inequality x < 3**

The following graph represents the inequality x < 3. Note that the open circle on 3 shows that 3 is not a solution to x < 3.

**Graph of the Inequality x > 3**

Here are the graphs of x > 3 and x ≥ 3, respectively

**Graph of the Inequality x ≥ 3**

**Graph of the Inequality x ≠ 3**

An inequality with a “≠" sign has a solution set which is all the real numbers except a single point. Thus, to graph an inequality with a " ≠ " sign, graph the entire line with one point removed. For example, the graph of x ≠ 3 looks like −

Graph the following inequality on the number line −

**$\frac{x}{3}$ ≤ 7**

**Step 1:**

$\frac{x}{3}$ ≤ 7; x ≤ (7 × 3); x ≤ 21

We first locate the point 21 on the number line.

**Step 2:**

We put a closed circle on 21 and draw a thick line towards left to denote the inequality x ≤ 21

Graph the following inequality on the number line −

**x > −32**

**Step 1:**

We first locate the point −32 on the number line.

**Step 2:**

We put a open circle on −32 and draw a thick line towards right to denote the inequality

x > −32

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