Expressing Answers in Interval Notation

Interval notation is a simplified form of writing the solution to an inequality or system of inequalities, using the bracket and parenthesis symbols in lieu of the inequality symbols. Intervals with parentheses are called open intervals, meaning the variable cannot have the value of the endpoints. For example, the solution 3 < x < 5 is written (3,5) in interval notation, because x cannot be equal to 3 or 5. Express your answers in interval notation by graphing the solution on a number line to determine the upper and lower bounds of the variable.

Step 1

Determine the values of the variable that make the inequality true. For example, the values of x that make the inequality 3x – 7 < 5 true are x < 4.

Step 2

Graph these values on the number line using open dots to represent < and > and closed dots to represent ≤ and ≥. In the above example, draw an open dot at the point corresponding to 4 on the number line and an arrow pointing to the left on the number line to indicate x < 4.

Step 3

Write the lower bound of the variable, with a left bracket “[” if the variable can have that value, or a left parenthesis “(” if it cannot or if the lower bound is negative infinity. In the example, the lower bound of x is negative infinity, so write “(-∞.”

Step 4

Write a comma after the lower bound and then write the upper bound of the variable, followed by a right bracket “]” if the variable can have that value, or a right parenthesis “)” if it cannot or if the upper bound is positive infinity. In the above example, the upper bound is 4 and x cannot have that value, so write “,4)”, making your answer in interval notation (-∞,4).

TL;DR (Too Long; Didn’t Read)

If there are other intervals of the variable, connect them with the union symbol “v.” Order the intervals from lowest to highest value. For example, if x ≥ 8 were another solution to the inequality in our example you would write (-∞,4) v [8,∞) as the interval.