0.8 Linear Inequalities
Figure 0.8.1: Several red winner’s ribbons lie on a white table.
It is not easy to make the honor roll at most top universities. Suppose students were required to carry a course load of at least 12 credit hours and maintain a grade point average of 3.5 or above. How could these honor roll requirements be expressed mathematically? In this section, we will explore various ways to express different sets of numbers and inequalities.
Absolute-value inequalities — those of the form \(|X| < k\), \(|X| \geq k\), and so on — are taken up alongside absolute-value functions in §1.6.
0.8.1 Using Interval Notation
Indicating the solution to an inequality such as \(x \geq 4\) can be achieved in several ways.
We can use a number line as shown in the figure below. The blue ray begins at \(x = 4\) and, as indicated by the arrowhead, continues to infinity, which illustrates that the solution set includes all real numbers greater than or equal to 4.
Figure 0.8.2: The set \(\{x : x \geq 4\}\), written \([4,\, \infty)\) in interval notation.
We can use set-builder notation: \(\left\{ x \middle| x \geq 4 \right\},\) which translates to “all real numbers \(x\) such that \(x\) is greater than or equal to 4.” Notice that braces are used to indicate a set.
The third method is interval notation, in which solution sets are indicated with parentheses or brackets. The solutions to \(x \geq 4\) are represented as \(\left\lbrack {4,\infty} \right).\) This is perhaps the most useful method, as it applies to concepts studied later in this course and to other higher-level math courses.
The main concept to remember is that parentheses represent solutions greater or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be “equaled.” A few examples of an interval, or a set of numbers in which a solution falls, are \(\left\lbrack {-2{,}6} \right),\) or all numbers between \(-2\) and \(6,\) including \(-2,\) but not including \(6;\) \(\left( {- 1{,}0} \right),\) all real numbers between, but not including \(-1\) and \(0;\) and \(\left( {- \infty,1} \right\rbrack,\) all real numbers less than and including \(1.\) Table 0.8.3 outlines the possibilities.
| Set Indicated | Interval Notation | Set-Builder Notation |
|---|---|---|
| All real numbers between \(a\) and \(b\), but not including \(a\) or \(b\) | \((a,b)\) | \(\{x \mid a < x < b\}\) | |
| All real numbers greater than \(a\), but not including \(a\) | \((a,\infty)\) | \(\{x \mid x > a\}\) | |
| All real numbers less than \(b\), but not including \(b\) | \((-\infty,b)\) | \(\{x \mid x < b\}\) | |
| All real numbers greater than \(a\), including \(a\) | \([a,\infty)\) | \(\{x \mid x \geq a\}\) | |
| All real numbers less than \(b\), including \(b\) | \((-\infty,b]\) | \(\{x \mid x \leq b\}\) | |
| All real numbers between \(a\) and \(b\), including \(a\) | \([a,b)\) | \(\{x \mid a \leq x < b\}\) | |
| All real numbers between \(a\) and \(b\), including \(b\) | \((a,b]\) | \(\{x \mid a < x \leq b\}\) | |
| All real numbers between \(a\) and \(b\), including \(a\) and \(b\) | \([a,b]\) | \(\{x \mid a \leq x \leq b\}\) | |
| All real numbers less than \(a\) or greater than \(b\) | \((-\infty,a) \cup (b,\infty)\) | \(\{x \mid x < a \text{ or } x > b\}\) | |
| All real numbers | \((-\infty,\infty)\) | \(\{x \mid x \text{ is all real numbers}\}\) |
Table 0.8.3: Possible interval notations for sets of real numbers.
0.8.2 Using the Properties of Inequalities
When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities. We can use the addition property and the multiplication property to help us solve them. The one exception is when we multiply or divide by a negative number; doing so reverses the inequality symbol.
0.8.3 Solving Inequalities in One Variable Algebraically
As the examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations; we combine like terms and perform operations. To solve, we isolate the variable.
0.8.4 Understanding Compound Inequalities
A compound inequality includes two inequalities in one statement. A statement such as \(4 < x \leq 6\) means \(4 < x\) and \(x \leq 6.\) There are two ways to solve compound inequalities: separating them into two separate inequalities or leaving the compound inequality intact and performing operations on all three parts at the same time. We will illustrate both methods.
Solve the compound inequality with variables in all three parts: \(3 + x > 7x - 2 > 5x - 10.\)
Solution
Let's try the first method. Write two inequalities:
\[\begin{array}{lll} {3 + x > 7x - 2} & {\quad \text{and}\qquad} & {7x - 2 > 5x - 10} \\ {\mspace{22mu} 3 > 6x - 2} & & {2x - 2 > -10} \\ {\mspace{22mu} 5 > 6x} & & {\quad 2x > -8} \\ {\mspace{22mu}\frac{5}{6} > x} & & {\quad x > -4} \\ {\mspace{23mu} x < \frac{5}{6}} & & {\mspace{27mu}-4 < x} \end{array}\]
The solution set is \(-4 < x < \frac{5}{6}\) or in interval notation \(\left( {\operatorname{}-4,\frac{5}{6}} \right).\) Notice that when we write the solution in interval notation, the smaller number comes first. We read intervals from left to right, as they appear on a number line. See the figure below.
Figure: The compound inequality \(-4 < x < \tfrac{5}{6}\) as a number line.
0.8.5 Summary
- Interval notation. Square brackets include endpoints; parentheses exclude them. \(\infty\) is not a number and always pairs with a parenthesis. Disjoint pieces join with \(\cup\).
- Inequality properties. Add or subtract the same quantity on both sides freely. Multiplying or dividing by a negative number reverses the inequality.
- Solving algebraically. Isolate the variable as in equations, watching for sign reversals.
- Compound inequalities. Split into two simultaneous inequalities, or operate on all three parts at once. The solution is an interval (or a union of intervals).
For absolute-value inequalities — those of the form \(|X| < k\), \(|X| \geq k\), and so on — see §1.6.