Mathematics » Equations and Inequalities » Solving Linear Inequalities

# Interval Notation

## Interval Notation

### Examples:

 $$(4;12)$$ Round brackets indicate that the number is not included. This interval includes all real numbers greater than but not equal to $$\text{4}$$ and less than but not equal to $$\text{12}$$. $$(-\infty ;-1)$$ Round brackets are always used for positive and negative infinity. This interval includes all real numbers less than, but not equal to $$-\text{1}$$. $$[1;13)$$ A square bracket indicates that the number is included. This interval includes all real numbers greater than or equal to $$\text{1}$$ and less than but not equal to $$\text{13}$$.

It is important to note that this notation can only be used to represent an interval of real numbers.

We represent the above answer in interval notation as $$(-\infty ; -\cfrac{1}{2}]$$

## Example

### Question

Solve for $$r$$:

$6 – r > 2$

Represent the answer on a number line and in interval notation.

### Rearrange and solve for $$r$$

\begin{align*} -r & > 2 – 6 \\ -r & > -4 \end{align*}

### Multiply by $$-\text{1}$$ and reverse inequality sign

$r < 4$

### Represent the answer in interval notation

$(-\infty ; 4)$

## Example

### Question

Solve for $$q$$:

$4q + 3 < 2(q + 3)$

Represent the answer on a number line and in interval notation.

### Expand the bracket

\begin{align*} 4q + 3 & < 2(q + 3) \\ 4q + 3 & < 2q + 6 \end{align*}

### Rearrange and solve for $$q$$

\begin{align*} 4q + 3 & < 2q + 6 \\ 4q – 2q & < 6 – 3 \\ 2q & < 3 \end{align*}

### Divide both sides by $$\text{2}$$

\begin{align*} 2q & < 3 \\ q & < \cfrac{3}{2} \end{align*}

### Represent the answer in interval notation

$$(-\infty ; \cfrac{3}{2})$$

## Example

### Question

Solve for $$x$$:

$5 \le x + 3 < 8$

Represent the answer on a number line and in interval notation.

### Subtract $$\text{3}$$ from all the parts of the inequality

$\begin{array}{ccccc} 5 – 3 & \le & x + 3 – 3 & < & 8 – 3 \\ 2 & \le & x & < & 5 \end{array}$

### Represent the answer in interval notation

$$[2 ; 5)$$