Intermediate Algebra 098A Chapter 9 Inequalities and Absolute Value

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Linear Inequalities – 3.2 Def: A linear inequality in one variable is an inequality that can be written in the form ax + b < 0 where a and b are real numbers and a is not equal to 0.

Solve by Graphing Graph the left and right sides and find the point of intersection Determine where x values are above and below. –Solution is x values – y is not critical

Example solve by graphing

Addition Property of Inequality If a < b, then a + c = b + c for all real numbers a, b, and c

Multiplication Property of Inequality For all real numbers a,b, and c If a 0, then ac < bc If a bc

Compound Inequalities 9.1 Def: Compound Inequality: Two inequalities joined by “and” or “or”

Intersection - Disjunction Intersection: For two sets A and B, the intersection of A and B, is a set containing only elements that are in both A and B.

Solving inequalities involving and 1. Solve each inequality in the compound inequality 2. The solution set will be the intersection of the individual solution sets.

Union - conjunction For two sets A and B, the union of A and B is a set containing every element in A or in B.

Solving inequalities involving “or” Solve each inequality in the compound inequality The solution set will be the union of the individual solution sets.

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Intermediate Algebra 098A Section 9.2 Absolute Value Equations

If |x|= a and a > 0, then x = a or x = -a If |x| = a and a < 0, the solution set is the empty set.

Procedure for Absolute Value equation |ax+b|=c 1. Isolate the absolute the absolute value. 2. Set up two equations ax + b = c ax + b = -c 3. Solve both equations 4. Check solutions

Procedure Absolute Value equations: |ax + b| = |cx + d| 1. Separate into two equations ax + b = cx + d ax + b = -(cx + d) 2. Solve both equations 3. Check solutions

Intermediate Algebra 098A Section 9.3 Absolute Value Inequalities

Inequalities involving absolute value |x| < a 1. Isolate the absolute value 2. Rewrite as two inequalities x -a) 3. Solve both inequalities 4. Intersect the two solutions note the use of the word “and” and so note in problem.

Sample Problem |5x +1| + 1 < 10 Answer [-2, 8/5]

Inequalities |x| > a 1. Isolate the absolute value 2. Rewrite as two inequalities x > a or –x > a (or x < -a) 3. Solve the two inequalities – union the two sets **** Note the use of the word “or” when writing problem.

Sample Problem

Answer

Intermediate Algebra 9.4 Graphing Linear Inequalities in Two Variables and Systems of Linear Inequalities

Def: Linear Inequality in 2 variables is an inequality that can be written in the form ax + by < c where a,b,c are real numbers. Use or >

Def: Solution & solution set of linear inequality Solution of a linear inequality in two variables is a pair of numbers (x,y) that makes the inequality true. Solution set is the set of all solutions of the inequality.

Procedure: graphing linear inequality 1. Set = and graph 2. Use dotted line if strict inequality or solid line if weak inequality 3. Pick point and test for truth –if a solution 4. Shade the appropriate region.

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Linear inequalities on calculator Set = Solve for Y Input in Y= Scroll left and scroll through icons and press [ENTER] Press [GRAPH]

Calculator Problem

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