Chapter 2Chapter 2

InequalitiesInequalities

Lesson 2-1 Graphing and Writing Lesson 2-1 Graphing and Writing InequalitiesInequalities

INEQUALITYINEQUALITY – a statement that two – a statement that two quantities are not equal.quantities are not equal.

SOLUTION OF AN INEQUALITYSOLUTION OF AN INEQUALITY – any value – any value that makes the inequality true (always that makes the inequality true (always test your solution with positive and test your solution with positive and negative numbers and zero to make sure negative numbers and zero to make sure it is true).it is true).

SOLID CIRCLESOLID CIRCLE – endpoint is a solution – endpoint is a solution ≥, ≤≥, ≤OPEN CIRCLEOPEN CIRCLE – endpoint not a solution – endpoint not a solution >,<>,<

Lesson 2-1 (cont.)Lesson 2-1 (cont.)

Hint: Hint: If you write an inequality with If you write an inequality with the variable on the left side of the the variable on the left side of the inequality symbol, the symbol points inequality symbol, the symbol points in the direction that you should in the direction that you should shade. shade.

y > 4 shades to the right.y > 4 shades to the right.

8 8 > x (first rewrite as x < 8, then > x (first rewrite as x < 8, then shade to the left)shade to the left)

Lesson 2-1 Graphing Lesson 2-1 Graphing InequalitiesInequalities

DISCDISC DDraw – draw a number line and a circle at raw – draw a number line and a circle at

the endpoint.the endpoint. IInclude? – look at the inequality sign to nclude? – look at the inequality sign to

determine whether the circle should be determine whether the circle should be solid or empty.solid or empty.

SShade - shade in the correct direction.hade - shade in the correct direction. CCheck – substitute a value on the solution heck – substitute a value on the solution

side into the expression to check that the side into the expression to check that the inequality is true.inequality is true.

Lesson 2-2 Solving Inequalities by Lesson 2-2 Solving Inequalities by Adding or SubtractingAdding or Subtracting

Solve an inequality as you would Solve an inequality as you would solve an equation. Always check the solve an equation. Always check the endpoint and the inequality symbol.endpoint and the inequality symbol.

Properties of inequality for addition Properties of inequality for addition and subtraction work just as and subtraction work just as properties of equality.properties of equality.

You may wish to rewrite your You may wish to rewrite your inequality with the variable on the inequality with the variable on the left if necessary.left if necessary.

Lesson 2-3 Solving Inequalities Lesson 2-3 Solving Inequalities by Multiplying and Dividingby Multiplying and Dividing

Use inverse operations to solve Use inverse operations to solve inequalitiesinequalities

5x < 20 5x < 20 Divide both sides by 5Divide both sides by 5

x < 4x < 4 x/2 > 9 x/2 > 9 Multiply both sides by 2Multiply both sides by 2

x > 18x > 18 Multiplication or division by a Multiplication or division by a negative negative

number reverses the inequality symbol.

Lesson 2-4 Multi-Step Lesson 2-4 Multi-Step InequalitiesInequalities

Simplify both sides before solvingSimplify both sides before solving

3x – 4 > -2 + 73x – 4 > -2 + 7

3x – 4 > 53x – 4 > 5

3x > 93x > 9

x > 3x > 3

Remember: The first step in solving is Remember: The first step in solving is to use the distributive property if to use the distributive property if possible. possible.

Lesson 2-5 Solving Inequalities Lesson 2-5 Solving Inequalities with Variables on both Sideswith Variables on both Sides

Collect variables on one side using Collect variables on one side using properties of inequality.properties of inequality.

2x + 9 2x + 9 >> x – 5 x – 5

x + 9 x + 9 >> -5 -5

x x >> -14 -14

Lesson 2-5 (cont)Lesson 2-5 (cont)

Identity: When solving an inequality, Identity: When solving an inequality, if you get a statement that is always if you get a statement that is always true, it is an identity, and all real true, it is an identity, and all real numbers are solutions. numbers are solutions.

EX: x + 9 > x + 4EX: x + 9 > x + 4

9 > 49 > 4

This statement is always true. All real This statement is always true. All real numbers are solutions.numbers are solutions.

Lesson 2-5 (cont)Lesson 2-5 (cont)

Contradiction: When solving an Contradiction: When solving an inequality, if you get a false inequality, if you get a false statement that is never true, it is a statement that is never true, it is a contraction and has no solutions.contraction and has no solutions.

EX: x + 8 < x + 3EX: x + 8 < x + 3

8 < 38 < 3

This statement is never true. There This statement is never true. There are no solutions.are no solutions.

Lesson 2-6 Solving Compound Lesson 2-6 Solving Compound InequalitiesInequalities

Compound InequalityCompound Inequality – Two simple – Two simple inequalities combined into one statement inequalities combined into one statement by the words AND or OR.by the words AND or OR.

IntersectionIntersection – The overlapping region of – The overlapping region of two inequalities joined by AND that shows two inequalities joined by AND that shows the numbers that are solutions of both the numbers that are solutions of both inequalities.inequalities.

UnionUnion – The combined regions of two – The combined regions of two inequalities joined by OR that shows the inequalities joined by OR that shows the numbers that are solutions of either.numbers that are solutions of either.

Lesson 2-6 (cont)Lesson 2-6 (cont)

6 < x + 4 6 < x + 4 << 9 9

6 < x + 4 AND x + 4 6 < x + 4 AND x + 4 << 9 9

2 < x AND x 2 < x AND x << 5 5

The solution is the INTERSECTION of The solution is the INTERSECTION of the two inequalities and can be the two inequalities and can be written:written:

2 < x 2 < x << 5 5

Lesson 2-6 (cont.)Lesson 2-6 (cont.)

x – 7 x – 7 >> -5 OR x + 2 < -1 -5 OR x + 2 < -1

x x >> 2 OR x < -3 2 OR x < -3

The solution is the UNION of the two The solution is the UNION of the two inequalities.inequalities.

Remember: Solving compound Remember: Solving compound inequalities requires solving two inequalities requires solving two separate inequalities.separate inequalities.

Lesson 2-7 Solving Absolute-Lesson 2-7 Solving Absolute-Value InequalitiesValue Inequalities

Absolute-Value Inequalities Absolute-Value Inequalities Involving <Involving < The inequality |x| < a (when a > 0) asks, The inequality |x| < a (when a > 0) asks,

“what values of x have an absolute value “what values of x have an absolute value less than a?” The solutions are the less than a?” The solutions are the numbers between a and –a.numbers between a and –a.

|x| < 5 “what values of x have an |x| < 5 “what values of x have an absolute value less than 5? -5 < x < 5absolute value less than 5? -5 < x < 5

(any number between -5 AND 5).(any number between -5 AND 5).

Lesson 2-6 (cont.)Lesson 2-6 (cont.)

Absolute-Value Inequalities Involving >Absolute-Value Inequalities Involving > The inequality |x| The inequality |x| >> a (when a > 0) asks, a (when a > 0) asks,

“what values of x have an absolute value “what values of x have an absolute value greater than a?” The solutions are the greater than a?” The solutions are the numbers less than –a OR greater than a.numbers less than –a OR greater than a. |x| |x| >> 5 “what values of x have an absolute 5 “what values of x have an absolute

value less than -5 or greater than 5?value less than -5 or greater than 5?

Any number less than -5 or greater than 5.Any number less than -5 or greater than 5.

Lesson 2-6 (cont.)Lesson 2-6 (cont.)Steps for Solving Absolute-Value Steps for Solving Absolute-Value

InequalitiesInequalities1. Isolate the absolute-value expression2. Write two cases – one positive and one negative3. Solve each case.

EX: 2|x + 2| ≥ 16 |x + 2| ≥ 8 1. Divide both sides by 2 to

isolate the absolute value expression.

x + 2 ≥ 8 x + 2 ≤ -8 2. Write the two cases. It is an

OR statement because it is ≥.

x ≥ 6 x ≤ -10 3. Solve each case.

X ≤ - 10 OR x ≥ 6