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SectionLinear Inequalities

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 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
  1. 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.

  2. We may multiply or divide both sides of an inequality by a positive number.

  3. If we multiply or divide both sides of an inequality by a negative number, we must reverse the direction of the inequality symbol.

Example36

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.

Example37

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.\\ -6 \amp{}\le{} \amp 3x\amp {}{}\amp {}\le{} 6\hphantom{blank} \amp\text{Divide 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.

SubsectionInterval Notation

The solutions of the inequality in Example37 form an interval. 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 Figure38a). 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 Figure38b).

closed interval and open interval
Figure38
Caution39

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 Figure40. 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.

infinite intervals
Figure40

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 Figure41, is the union of two intervals and is denoted by \((-\infty,-2) \cup (2,\infty)\text{.}\)

number line with two disjoint infinite intervals
Figure41

Many solutions of inequalities are intervals or unions of intervals.

Example42

Write each of the solution sets with interval notation and graph the solution set on a number line.

  1. \(3 \le x \lt 6\)

  2. \(x \ge -9\)

  3. \(x\le 1 ~\text{ or }~ x\gt 4\)

  4. \(-8 \lt x \le -5 ~\text{ or }~ -1 \le x \lt 3\)

Solution
  1. \([3, 6)\text{.}\) This is called a half-open or half-closed interval. (See Figure43.)

    half-open interval
    Figure43
  2. \([-9,\infty)\text{.}\) 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 Figure44.)

    half-closed infinite interval
    Figure44
  3. \((-\infty, 1] \cup (4, \infty)\text{.}\) The word or describes the union of two sets. (See Figure45.)

    disjoint infinite intervals
    Figure45
  4. \((-8,-5] \cup [-1, 3)\text{.}\) (See Figure46.)

    disjoint finite intervals
    Figure46