# Solution Sets

## Solution Sets for Equations

The set containing all the solutions of an equation is called the solution set for that equation.

If an equation has no solutions, we write $\varnothing$ for the solution set. $\varnothing$ means the null set (or empty set).

 Equation Solution Set $3x+5=11$ $\left\{2\right\}$ ${x}^{2}=x$ $\left\{0,1\right\}$ $x+1=1+x$ $\text{R}$ (the set of all real numbers) $x+1=x$ $\varnothing$ (the empty set)

Sometimes, you may be given a replacement set, and asked to test whether the equation is true for all values in the replacement set.

Example 1:

Find the solution set for the equation $z+z=z×z$ if the replacement set is $\left\{0,1,2,3,\right\}$ .

One method of solving this problem is to test all the values in the replacement set using a table.

$\begin{array}{|ccc|}\hline z& z+z=z×z& \text{Result}\\ 0& \begin{array}{l}0+0\stackrel{?}{=}0×0\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0=0\end{array}& \\ 1& \begin{array}{l}1+1\stackrel{?}{=}1×1\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}2\ne 1\end{array}& \\ 2& \begin{array}{l}2+2\stackrel{?}{=}2×2\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}4=4\end{array}& \\ 3& \begin{array}{l}3+3\stackrel{?}{=}3×3\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}6\ne 9\end{array}& \\ \hline\end{array}$

So, the solution set for this equation is $\left\{0,2\right\}$ .

## Solution Sets for Inequalities

Solution sets for inequalities are often infinite sets; we can't list all the numbers. So, we use a special notation.

Example 2:

Solve the inequality

$x+2>-3$ .

By subtracting 2 from both sides, we get the equivalent inequality

$x>-5$ .

So, the solution set is

$\left\{x\text{\hspace{0.17em}}|\text{\hspace{0.17em}}x>-5\right\}$ .

(To read this, you would say: " $x$ such that $x$ is greater than negative five." The | symbol means "such that" in this case.)

Often, the solutions to inequalities are also written in interval notation .