Algebra 1 Section 4.1

Inequalities An inequality is a mathematical sentence stating that two quantities are not always equal. There are five basic inequality symbols.

is greater than or equal to Inequality Symbols < > ≠ ≤ ≥ is less than is greater than is not equal to is less than or equal to is greater than or equal to 4 < 6 5 > 3 -2 ≠ 4 -1 ≤ 8 6 ≥ 0

Trichotomy Property If a and b represent two real numbers, then a = b, a > b, or a < b.

Inequalities An inequality can be negated by putting a slash through it. For example, x > 5 is read, “x is not greater than 5.” This is equivalent to x ≤ 5.

Example 1 State an inequality statement equivalent to x ≤ y. Answer: x > y.

Inequalities Any value of the variable that makes an inequality true is a solution of the inequality. There are typically an infinite number of solutions to each inequality.

Example 2 a + 3 > 6 2 + 3 > 6 is false. 3 + 3 > 6 is false. 4 + 3 > 6 is true. 4 is a solution.

Example 2 b. 6 ≥ 2b 6 ≥ 2(2) is true. 6 ≥ 2(3) is true. 6 ≥ 2(4) is false. 2 and 3 are solutions.

Example 2 1 – c < -2 1 – c ≥ -2 1 – 2 ≥ -2 is true. 1 – 4 ≥ -2 is false. 2 and 3 are solutions.

Graphing Inequalities It is usually not possible to list all the solutions to an inequality. Therefore, we make a graph on a number line.

Graphing Inequalities A circle is used when the number itself is not part of the solution set. 5 > x is usually rewritten: x < 5

Example 3 Graph x ≥ -3. Since the inequality symbol includes equal to, a solid dot is placed at -3.

Example 4 Graph x ≠ 2. x > 2 or x < 2 Any number greater or less than 2 is a solution.

Example 6 Let c = price of an item c ≥ 9.99 (b) Let p = number of people p ≤ 225

Homework: pp. 146-148