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2.6 Inequalities
De nition 6.1. An inequality is a statement that two quantities or expressions are not equal. An inequality can be expressed using one of the relations <, ≤, >, ≥.
relation reads as inequality example reads as < less than a < b a is less than b ≤ less than or equal to a ≤ b a is less than or equal to b
greater than a > b a is greater than b ≥ greater than or equal to a ≥ b a is lgreater than or equal to b
De nition 6.2. Solving an inequality that involves a variable x means nding all of its solutions. A solution for a inequality in a variable x is a number b that yields a true statement when substituted for x. Equivalent inequalities are inequalities that have exactly the same solutions.
Example 6.3. Consider the inequality
2 x + 3 > 11. We have that the number 5 is a solution for the given inequality, but 1 is not, since x 2 x + 3 > 11 conclusion 1 5 > 11 false statement 5 13 > 11 true statement
Most inequalities have an in nite number of solutions. For example the solutions of the inequality 2 < x < 5 consist of every real number between 2 and 5. We call this set of numbers an open interval and denote it by (2, 5). Intervals
1
Methods for solving inequalities. The methods for solving inequalities in x are similar to those used for solving equations, with one exception. When multiplying or dividing an inequality by a negative number we have to reverse the inequality.
Properties of inequalities:
- we can add or subtract a real number to both sides of an inequality and we obtain an equivalent inequality
- we can multiply or divide both sides of an inequality by a positive real number and we obtain an equivalent inequality (1) If a < b, then a + c < b + c and a − c < b − c.
(2) If a < b and c > 0 , then a · c < b · c and
a c
b c
(3) If a < b and c < 0 , then a · c > b · c and
a c
b c
Remark 6.4. We cannot multiply both sides by an expression cotaining the variable x if we cannot say that the expression is either always positive or always negative.
Example 6.5. Solve the inequality 5 x − 7 ≤ 3.
Solution. Given the properties of inequality we obtain the following equivalent inequali- ties: 5 x − 7 ≤ 3 given
(5x − 7) + 7 ≤ 3 + 7 add 7
5 x ≤ 10 simplify
5 x 5
divide by 5 (positive)
x ≤ 2 simplify
So the solutions to the given inequality are all numbers such that x ≤ 2. This is the interval (−∞, 2].
Properties of absolute value. The equation |x| < 3 is equivalent to − 3 < x < 3 , thus its solutions consist of the interval (− 3 , 3).
In general the equation |x| < a, where a > 0 , is equivalent to −a < x < a, thus its solutions consist of the interval (−a, a), and the equation |x| > a, where a > 0 , is equivalent to x < −a or x > a, thus its solutions consist of the union of intervals (−∞, a) ∪ (a, ∞).
Example 6.9. Solving an inequality containing an absolute value. Solve the inequality and express the solutions in terms of intervals: |x − 3 | < 1.
Solution.
|x − 3 | < 1 given − 1 < x − 3 < 1 property of absolute value −1 + 3 < x < 1 + 3 isolate x by adding 3 2 < x < 4 divide by 2 (positive)
Thus the solutions are the real numbers in the open interval (2, 4).
Example 6.10. Solving an inequality containing an absolute value. Solve the inequality and express the solutions in terms of intervals: | 2 x + 3| > 9.
Solution.
| 2 x + 3| > 9 given 2 x + 3 < − 9 or 2 x + 3 > 9 property of absolute value 2 x < − 12 or 2 x > 6 subtract 3 x < − 6 or x > 3 divide by 2 (positive)
Thus the solutions are the real numbers in the union of intervals (−∞, −6) ∪ (3, ∞).
Example 6.11. Solving a rational inequality (more complicated examples of these, using a sign diagram, in section 2.7).
Solve the inequality and express the solutions in terms of intervals:
x − 3
Solution. Since the numerator equals 5 which is a positive number, for the fraction to be positive, we need that the denominator, x − 3 , is also positive. Thus x − 3 > 0 or, equivalently, x > 3 , and the solutions are all numbers in the interval (3, ∞).
