**Introduction to Coordinates in One Dimension**

We envision the real numbers as laid out on a line, and we locate real numbers from left to right on this line. If *a < b* are real numbers then *a* will lie to the left of *b* on this line. See Fig. 1.1.

**Examples**

**Example 1**

On a real number line, plot the numbers −4, −1, 2, 6. Also plot the sets *S* = { *x* : − 8 ≤ *x* < −5} and *T* = {t : 7 < *t* ≤ 9}. Label the plots.

**Solution 1**

Figure 1.2 exhibits the indicated points and the two sets. These sets are called *half-open intervals* because each set includes one endpoint and not the other.

**Math Note:** The notation *S* = { *x* : − 8 ≤ *x* < −5} is called *set builder notation* . It says that *S* is the set of all numbers *x* such that *x* is greater than or equal to −8 and less than 5. We will use set builder notation throughout the book.

If an interval contains both its endpoints, then it is called a *closed interval* . If an interval omits both its endpoints, then it is called an *open interval* . See Fig. 1.3.

**Examples**

**Example 2**

Find the set of points that satisfy *x* − 2 < 4 and exhibit it on a number line.

**Solution 2 **

We solve the inequality to obtain *x* < 6. The set of points satisfying this inequality is exhibited in Fig. 1.4 .

**Example 3**

Find the set of points that satisfy the condition

and exhibit it on a number line.

**Solution 3 **

In case *x* + 3 ≥ 0 then ∣ *x* + 3∣ = *x* + 3 and we may write condition (*) as

*x* + 3 ≤ 2

or

*x* ≤ −1.

Combining *x* + 3 ≥ 0 and *x* ≤ −1 gives −3 ≤ *x* ≤ −1.

On the other hand, if *x* + 3 < 0 then ∣ *x* + 3∣ = −( *x* + 3). We may then write condition (*) as

−( *x* + 3) ≤ 2

or

−5 ≤ *x* .

Combining *x* + 3 < 0 and −5 ≤ *x* gives −5 ≤ *x* < −3.

We have found that our inequality ∣ *x* + 3∣ ≤ 2 is true precisely when either −3 ≤ *x* ≤ −1 or −5 ≤ *x* < −3. Putting these together yields −5 ≤ *x* ≤ −1. We display this set in Fig. 1.5 .

**You Try It:** Solve the inequality ∣ *x* −4∣ > 1. Exhibit your answer on a number line.

**You Try It:** On a real number line, sketch the set { *x* : *x* ^{2} − 1 < 3}.

Practice problems for this concept can be found at: Calculus Basics Practice Test.

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