Contents: This page corresponds to § 2.5 (p. 216) of the text. Show
Suggested Problems from Text: p. 225 #11, 12, 13, 14, 16, 28, 33, 35, 38, 41, 53, 56, 62, 63, 68, 69
Linear InequalitiesAn inequality is a comparison of expressions by either "less than" (<), "less than or equal to" (<=), "greater than" (>), or "greater than or equal to" (>=). Note that Html does not support the standard symbols for "less than or equal to" and "greater than or equal to", so we use <= and >= for these relations. Example 1. x + 3 <= 10 A solution for an inequality in x is a number such that when we substitute that number for x we have a true statement. So, 4 is a solution for example 1, while 8 is not. The solution set of an inequality is the set of all solutions. Typically an inequality has infinitely many solutions and the solution set is easily described using interval notation. The solution set of example 1 is the set of all x <= 7. In interval notation this set is (-inf, 7], where we use inf to stand for infinity. A linear inequality is one such that if we replaced the inequality with the equals relation, then we would have a linear equation. Solving linear inequalities is very much like solving linear equations, with one important difference.
You can see this using an inequality with no variables. Example 2.
Note: In general we may not multiply or divide both sides of an inequality by an expression with a variable, because some values of the variable may make the expression positive and some may make it negative. Example 3.
Look at the graphs of the functions on either side of the inequality. To satisfy the inequality, 7 - 2x needs to be less than 3. So we are looking for numbers x such that the point on the graph of y = 7 - 2x is below the point on the graph of y = 3. This is true for x > 2. In interval notation the solution set is (2, inf). There is another way to use a graphing utility to solve this inequality. In the Java Grapher the expression (7-2*x)L3 has the value 1 for numbers x that satisfy the inequality, and the value 0 for other numbers x. The picture below shows the graph of (7-2*x)L3 as drawn by the Grapher. Exercise 1:
Return to Contents Combinations of InequalitiesExample 4.
Example 5.
Exercise 2:
Return to Contents Inequalities Involving Absolute ValuesInequalities involving absolute values can be rewritten as combinations of inequalities. Let a be a positive number.
To make sense of these statements, think about a number line. The absolute value of a number is the distance the number is from 0 on the number line. So the inequality |x| < a is satisfied by numbers whose distance from 0 is less than a. This is the set of numbers between -a and a. The inequality |x| > a is satisfied by numbers whose distance from 0 is larger than a. This means numbers that are either larger than a, or less than -a. Example 6.
Example 7.
Exercise 3:
Return to Contents Polynomial InequalitiesExample 8.
Common Mistake
Example 9.
Exercise 4:
Return to Contents Rational InequalitiesA rational expression is one of the form polynomial divided by polynomial. In general, graphs of rational functions do have breaks. They are not defined at the zeros of the denominator. These are the only places where there are breaks, so we can use the same technique to solve rational inequalities that we use for polynomial inequalities. Example 10.
There are two important points to keep in mind when working with inequalities:
Exercise 5:
Return to Contents Does an inequality equation only have one correct solution?An inequality can have only one solution (that is, a single real value of x as the only solution). This can happen with any of the above-mentioned inequalities: absolute, compound, and quadratic. Let's look at some examples, starting with absolute value inequalities with one solution.
Can you have more than one solution in an inequality?An inequality can have more than one solution. The set of all solutions of an inequality is called the solution set. The symbol ≥/ means “is not greater than or equal to.” Value of x x + 5 ≥ −2 Is the inequality true?
Can an inequality have no solution?Note: Can an inequality have no solution? You bet it can! An inequality has no solution will always give you a contradiction, no matter what value you plug in for the variable.
What is the solution of inequality?To solve an inequality, isolate the variable on one side with all other constants on the other side. To accomplish this, perform opposite operations to manipulate the inequality. First, isolate the x by multiplying each side by two. Whatever you do to one side you must also do to the other side.
What are the rules of solving inequalities?When solving an inequality: • you can add the same quantity to each side • you can subtract the same quantity from each side • you can multiply or divide each side by the same positive quantity If you multiply or divide each side by a negative quantity, the inequality symbol must be reversed.
Are there an infinite amount of solutions to an inequality?Inequalities that produce a false statement have no real solutions. Inequalities that hold true for all values have infinite solutions.
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