To solve an equation we have to find the one (or many) values of $x$ which satisfy the equation. The solution set for an equation consists of a discrete set of values. For example, the solutions to $(x-3)^2=4$ are $x=1$ and $x=5$.

In this section, we will learn how to solve equations which involve inequalities. The solution to an inequality is usually an entire range of numbers. For example the inequality $(x-3)^2 \leq 4$ is equivalent to asking the question “for which values of $x$ is $(x-3)^2$ less than or equal to $4$.” The answer is the interval $[1,5] \equiv \{ x\in \mathbb{R}\ | \ 1 \leq x \leq 5 \}$.

The techniques used to deal with inequalities are roughly the same as the techniques which we learned for dealing with equations: we have to perform simplifying steps to both sides of the inequality until we obtain the answer.

The different type of inequality conditions are:

• $f(x) < g(x)$: a strict inequality. The function $f$ is always strictly less than $g$.
• $f(x) \leq g(x)$: the function $f$ is less than or equal to the function $g$.
• $f(x) > g(x)$: $f$ is strictly greater than $g$.
• $f(x) \geq g(x)$: $f$ is greater than or equal to $g$.

The solutions to an inequality correspond to subsets of the real line. Depending on the type of inequality we are dealing with, the answer will be either a closed or open interval:

• $[a,b]$: the closed interval from $a$ to $b$. This corresponds to the set of numbers between $a$ and $b$ on the real line, including the endpoints $a$ and $b$. $[a,b] = \{ x\in \mathbb{R}\ | \ a \leq x \leq b \}$.
• $(a,b)$: the open interval from $a$ to $b$. This corresponds to the set of numbers between $a$ and $b$ on the real line, not including the $a$ and $b$. $(a,b) = \{ x\in \mathbb{R}\ | \ a < x < b \}$.
• $[a,b)$: the mixed interval which includes the left endpoint $a$, but not the right endpoint $b$.

Sometimes the we will have to deal with intervals which consists of two disjoint parts:

• $[a,b] \cup [c,d]$: The set of all numbers that are either between $a$ and $b$ (inclusive) or between $c$ and $d$ (inclusive).

The main idea for solving inequalities is the same as solving equations except for one small special step. When multiplying by a negative number on both sides, the direction of the inequality must be flipped: $$f(x) \leq g(x) \qquad \Rightarrow \qquad -f(x) \geq -g(x).$$

To solve $(x-3)^2\leq 4$ we must dig towards the $x$ and undo all the operations that stand in our way: $$\begin{align*} & \ (x-3)^2 \leq 4, \nl -2 \leq & \ (x-3) \leq 2, \nl 1 \leq & \ \ \ \ \ x \ \ \ \ \leq 5. \end{align*}$$ where in the first step we took the square root operation (the inverse of the quadratic function) and then we added $3$ to both sides. The final answer is $x\in[1,5]$.

As you can see, solving inequalities is not more complicated than solving equations. Indeed, the best way to think about an inequality is in terms of the end points – which correspond to the equality condition.