Chapter 2 Inequalities

Lesson 2-1 Graphing and Writing Inequalities INEQUALITY – a statement that two quantities are not equal. SOLUTION OF AN INEQUALITY – any value that makes the inequality true (always test your solution with positive and negative numbers and zero to make sure it is true). SOLID CIRCLE – endpoint is a solution ≥, ≤ OPEN CIRCLE – endpoint not a solution >,,<

Lesson 2-1 (cont.) Hint: If you write an inequality with the variable on the left side of the inequality symbol, the symbol points in the direction that you should shade. y > 4 shades to the right. 8 > x (first rewrite as x x (first rewrite as x < 8, then shade to the left)

Lesson 2-1 Graphing Inequalities DISC Draw – draw a number line and a circle at the endpoint. Draw – draw a number line and a circle at the endpoint. Include? – look at the inequality sign to determine whether the circle should be solid or empty. Include? – look at the inequality sign to determine whether the circle should be solid or empty. Shade - shade in the correct direction. Shade - shade in the correct direction. Check – substitute a value on the solution side into the expression to check that the inequality is true. Check – substitute a value on the solution side into the expression to check that the inequality is true.

Lesson 2-2 Solving Inequalities by Adding or Subtracting Solve an inequality as you would solve an equation. Always check the endpoint and the inequality symbol. Solve an inequality as you would solve an equation. Always check the endpoint and the inequality symbol. Properties of inequality for addition and subtraction work just as properties of equality. Properties of inequality for addition and subtraction work just as properties of equality. You may wish to rewrite your inequality with the variable on the left if necessary. You may wish to rewrite your inequality with the variable on the left if necessary.

Lesson 2-3 Solving Inequalities by Multiplying and Dividing Use inverse operations to solve inequalities Use inverse operations to solve inequalities 5x < 20 Divide both sides by 5 5x < 20 Divide both sides by 5 x < 4 x < 4 x/2 > 9 Multiply both sides by 2 x/2 > 9 Multiply both sides by 2 x > 18 x > 18 Multiplication or division by a negative Multiplication or division by a negative number reverses the inequality symbol.

Lesson 2-4 Multi-Step Inequalities Simplify both sides before solving Simplify both sides before solving 3x – 4 > x – 4 > x – 4 > 5 3x – 4 > 5 3x > 9 3x > 9 x > 3 x > 3 Remember: The first step in solving is to use the distributive property if possible.

Lesson 2-5 Solving Inequalities with Variables on both Sides Collect variables on one side using properties of inequality. Collect variables on one side using properties of inequality. 2x + 9 > x – 5 2x + 9 > x – 5 x + 9 > -5 x + 9 > -5 x > -14 x > -14

Lesson 2-5 (cont) Identity: When solving an inequality, if you get a statement that is always true, it is an identity, and all real numbers are solutions. Identity: When solving an inequality, if you get a statement that is always true, it is an identity, and all real numbers are solutions. EX: x + 9 > x > 4 9 > 4 This statement is always true. All real numbers are solutions.

Lesson 2-5 (cont) Contradiction: When solving an inequality, if you get a false statement that is never true, it is a contraction and has no solutions. Contradiction: When solving an inequality, if you get a false statement that is never true, it is a contraction and has no solutions. EX: x + 8 < x < 3 8 < 3 This statement is never true. There are no solutions.

Lesson 2-6 Solving Compound Inequalities Compound Inequality – Two simple inequalities combined into one statement by the words AND or OR. Compound Inequality – Two simple inequalities combined into one statement by the words AND or OR. Intersection – The overlapping region of two inequalities joined by AND that shows the numbers that are solutions of both inequalities. Intersection – The overlapping region of two inequalities joined by AND that shows the numbers that are solutions of both inequalities. Union – The combined regions of two inequalities joined by OR that shows the numbers that are solutions of either. Union – The combined regions of two inequalities joined by OR that shows the numbers that are solutions of either.

Lesson 2-6 (cont) 6 < x + 4 < 9 6 < x + 4 < 9 6 < x + 4 AND x + 4 < 9 2 < x AND x < 5 The solution is the INTERSECTION of the two inequalities and can be written: 2 < x < 5 2 < x < 5

Lesson 2-6 (cont.) x – 7 > -5 OR x OR x + 2 < -1 x > 2 OR x 2 OR x < -3 The solution is the UNION of the two inequalities. Remember: Solving compound inequalities requires solving two separate inequalities.

Lesson 2-7 Solving Absolute- Value Inequalities Absolute-Value Inequalities Involving < Absolute-Value Inequalities Involving < The inequality |x| 0) asks, “what values of x have an absolute value less than a?” The solutions are the numbers between a and –a. The inequality |x| 0) asks, “what values of x have an absolute value less than a?” The solutions are the numbers between a and –a. |x| < 5 “what values of x have an absolute value less than 5? -5 < x < 5 |x| < 5 “what values of x have an absolute value less than 5? -5 < x < 5 (any number between -5 AND 5).

Lesson 2-6 (cont.) Absolute-Value Inequalities Involving > Absolute-Value Inequalities Involving > The inequality |x| > a (when a > 0) asks, “what values of x have an absolute value greater than a?” The solutions are the numbers less than –a OR greater than a. The inequality |x| > a (when a > 0) asks, “what values of x have an absolute value greater than a?” The solutions are the numbers less than –a OR greater than a. |x| > 5 “what values of x have an absolute value less than -5 or greater than 5? |x| > 5 “what values of x have an absolute 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.) Steps for Solving Absolute-Value Inequalities 1.Isolate the absolute-value expression 2.Write two cases – one positive and one negative 3.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 ≤ -82. Write the two cases. It is an OR statement because it is ≥. x ≥ 6 x ≤ Solve each case. X ≤ - 10 OR x ≥ 6