Equations and Inequalities Chapter 2 Equations and Inequalities

Chapter Sections 2.1 – Solving Linear Equations 2.2 – Problem Solving and Using Formulas 2.3 – Applications of Algebra 2.4 – Additional Application Problems 2.5 – Solving Linear Inequalities 2.6 – Solving Equations and Inequalities Containing Absolute Values Chapter 1 Outline

Solving Linear Inequalities § 2.5 Solving Linear Inequalities

Solve Inequalities An inequality is a mathematical statement containing one or more inequality sign(<, >, ,  ).

Properties Used to Solve Inequalities For all real numbers a, b, and c: If a > b, then a + c > b + c. If a > b, then a – c > b – c. If a > b and c > 0 then ac > bc. 4. If a > b and c > 0 then . If a > b and c < 0 then ac < bc. 6. If a > b and c < 0 then .

Solve the Inequality The solution set is {x|x ≥ -2}. Any real number greater than or equal to -2 will satisfy the inequality.

Graphing Intervals x  -1 -5 -4 -3 -2 -1 1 2 3 4 5 An arrow is used to show that the interval does not end. Endpoints are used to show the end of an interval. A closed circle is used to show that the endpoint is included in the answer. The symbols  and  will use this type of endpoint. An open circle is used to show that the endpoint is NOT included in the answer. The symbols > and < will use this type of endpoint.

Graphing Intervals x < 3 -2 < x < 0 -1.5  x  3 -5 -4 -3 -2 1 2 3 4 5 -2 < x < 0 -5 -4 -3 -2 -1 1 2 3 4 5 -1.5  x  3 -5 -4 -3 -2 -1 1 2 3 4 5

Solving Inequalities Example: Solve the inequality and graph the solution.

Solving Inequalities Example continued: Since -1 is always less than or equal to 14, the inequality is true for all real numbers.

Compound Inequalities A compound inequality is formed by joining two inequalities with the word and or or. Examples: 3 < x and x < 5 c  2 and c > -3 x < 2 and x > 4 5x – 3  7 or –x + 3 < -5

Solve Compound Inequalities Involving And The solution of a compound inequality using the word and is all the numbers that make both parts of the inequality true. This is the intersection of the solution sets of the two inequalities. Example: 3 < x and x < 5 Find the numbers that satisfy both inequalities. The solution set is the intersection of the two inequalities.

Solve Compound Inequalities Involving And Example: Solve x + 5 ≤ 8 and 2x – 9 > 7. To find the solution algebraically, begin by solving each inequality separately. and

Solve Compound Inequalities Involving And Example continued… Now take the intersection of the sets {x|x ≤ 3} and {x|x > 1}.

Solve Compound Inequalities Involving Or The solution to a compound inequality using the word or is all the numbers that make either of the inequalities a true statement. This is the union of the solution sets of the two inequalities. Example: x > 3 or x < 5 Find the numbers that satisfy at least one of the inequalities. The solution set is the intersection of the two inequalities.