In this section, you will...
solve linear inequalities
represent solutions to inequalities using a variety of representations
SectionLinear Inequalities
In the previous section, we talked about linear equations, which are mathematical statements that indicate that two expressions are equal. But what if we would like to compare expressions that are different "sizes"?
Supplemental Videos
The main topics of this section are also presented in the following videos:
SubsectionLinear Inequalities
The symbol \(\gt\) is called an inequality symbol, and the statement \(a\gt b\) is called an inequality. There are four inequality symbols:
\begin{align*}
\amp\gt \amp\amp\text{is greater than}\\
\amp\lt \amp\amp\text{is less than}\\
\amp\ge \amp\amp\text{is greater than or equal to}\\
\amp\le \amp\amp\text{is less than or equal to}
\end{align*}
Inequalities that include the symbols \(\gt\) or \(\lt\) are called strict inequalities; those that include \(\ge\) or \(\le\) are called nonstrict.
If an inequality has one or more variables in it, a solution of that inequality is any set of values that can replace the variables to produce a true statement. For example, \(3\) is a solution for \(2x>2\) since \(2\cdot3>2\) is a true statement. However, \(3\) is not the only solution! In fact, any number greater than \(1\) is a solution. While linear equations have exactly one, zero, or infinitely many solutions, linear inequalities can have much more complex solution sets.
Solving linear inequalities is very similar to solving linear equalities. The main difference is in multiplying and dividing: if we multiply or divide both sides of an inequality by a negative number, the direction of the inequality must be reversed. For example, if we multiply both sides of the inequality
\begin{equation*}
2\lt 5
\end{equation*}
by \(-3\text{,}\) we get
\begin{align*}
\alert{-3}(2) \amp\gt \alert{-3}(5)\amp\amp \text{Change inequality symbol from }\lt \text{ to }\gt.\\
-6 \amp\gt -15.
\end{align*}
Because of this property, the rules for solving linear equations must be revised slightly for solving linear inequalities.
To Solve a Linear Inequality
We may add the same number to both sides of an inequality or subtract the same number from both sides of an inequality without changing its solutions.
We may multiply or divide both sides of an inequality by a positive number.
If we multiply or divide both sides of an inequality by a negative number, we must reverse the direction of the inequality symbol.
Example53
Solve the inequality \(4 - 3x \ge -17\text{.}\)
Solution
Use the rules above to isolate \(x\) on one side of the inequality.
\begin{align*}
4 - 3x \amp\ge -17\amp\amp\text{Subtract 4 from both sides.}\\
-3x \amp\ge -21\amp\amp\text{Divide both sides by }-3.\\
x \amp\le 7
\end{align*}
Notice that we reversed the direction of the inequality when we divided by \(-3\text{.}\) Any number less than or equal to \(7\) is a solution of the inequality.
A compound inequality involves two inequality symbols. To solve a compound inequality, we use the same steps as before, applying the operations on all three "sides" of the inequality symbols.
Example54
Solve \(4 \le 3x + 10 \le 16\text{.}\)
Solution
We isolate \(x\) by performing the same operations on all three sides of the inequality.
\begin{alignat*}{3}
4 \amp{}\le{} 3x\amp {}+{}\amp 10 \amp{}\le{} 16\hphantom{blank} \amp\text{Subtract }10\text{ from each piece.}\\
-6 \amp{}\le{} \amp 3x\amp {}{}\amp {}\le{} 6\hphantom{blank} \amp\text{Divide each piece by }3.\\
-2 \amp{}\le{} \amp x\amp {}{}\amp{}\le{} 2\hphantom{blank}\amp
\end{alignat*}
The solutions are all numbers between \(-2\) and \(2\text{,}\) inclusive.
Caution55
Notice in the previous example that in a compound inequality, both of the inequality symbols are in the same "direction". You typically would not see something like \(4\lt x \geq 8\) or \(-3\geq x \leq 2\text{.}\)
SubsectionInterval Notation
A common way to represent solutions of an inequality is with interval notation. An interval is a set that consists of all the real numbers between two numbers \(a\) and \(b\text{.}\)
The set \(-2 \le x \le 2\) includes its endpoints \(-2\) and \(2\text{,}\) so we call it a closed interval, and we denote it by \([-2, 2]\) (see Figure56a). The square brackets tell us that the endpoints are included in the interval. An interval that does not include its endpoints, such as \(-2 \lt x \lt 2\text{,}\) is called an open interval, and we denote it with round brackets, \((-2, 2)\) (see Figure56b).
Caution57
Do not confuse the open interval \((-2, 2)\) with the point \((-2, 2)\text{!}\) The notation is the same, so you must decide from the context whether an interval or a point is being discussed.
We can also discuss infinite intervals, such as \(x\lt 3\) and \(x\ge -1\text{,}\) shown in Figure58. We denote the interval \(x\lt 3\) by \((-\infty, 3)\text{,}\) and the interval \(x\ge -1\) by \([-1, \infty)\text{.}\) The symbol \(\infty\text{,}\) for infinity, does not represent a specific real number but rather indicates that the interval continues forever along the real line. We always use round brackets next to \(\pm\infty\) in infinite intervals.
Finally, we can combine two or more intervals into a larger set. For example, the set consisting of \(x\lt -1\) or \(x\gt 2\text{,}\) shown in Figure59, is the union of two intervals and is denoted by \((-\infty,-2) \cup (2,\infty)\text{.}\)
Many solutions of inequalities are intervals or unions of intervals.
Example60
Write each of the solution sets with interval notation and graph the solution set on a number line.
\(3 \le x \lt 6\)
\(x \ge -9\)
\(x\le 1 ~\text{ or }~ x\gt 4\)
\(-8 \lt x \le -5 ~\text{ or }~ -1 \le x \lt 3\)
Solution
-
\([3, 6)\text{.}\) This is called a half-open or half-closed interval. (See Figure61.)
Figure61 -
\([-9,\infty)\text{.}\) We always use round brackets next to the symbol \(\infty\) because \(\infty\) is not a specific number and is not included in the set. (See Figure62.)
Figure62 -
\((-\infty, 1] \cup (4, \infty)\text{.}\) The word or describes the union of two sets. (See Figure63.)
Figure63 -
\((-8,-5] \cup [-1, 3)\text{.}\) (See Figure64.)
Figure64