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 on a number line

6de091a60b043b894bcacc16e0ee6a9c.png

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 on a number line

aa98069629d3597b02b485146a4831c4.png

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 on a number line

bf0d0956dfd72128df760123bc6fa414.png

Represent the answer in interval notation

\([2 ; 5)\)

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