Standard: M11.D.2.1.1 Algebra 3 Lesson 4.1 Objective: SSBAT solve and graph inequalities and compound inequalities on a number line. Standard: M11.D.2.1.1

iNEQUALITY A mathematical statement that involves any of the following symbols: <, >, ≤, ≥, or ≠

How is an inequality different from an equation? An equation such as 2x = 10 only has one solution. There is only one number that will work for x That number is x = 5 An inequality, such as 2x < 10, has many solutions. 4 is a solution because 2(4) = 8 which is less than 10 -2 is also a solution because 2(-2) = -4 which is less than 10 Any number less than 5 will be a solution

Solving Inequalities -3x > 15 x < -5 3x > -15 x > -5  Solve just like equations BUT if you Multiply or Divide by a negative number you SWITCH the direction of the inequality symbol. -3x > 15 x < -5 3x > -15 x > -5 Switch the sign. Do NOT switch the sign, did not divide by a negative

Graphing Solutions on a Number Line If < or > use an open dot You don’t want to include that point. If ≤ or ≥ use a closed dot You want to include the point because the solution is also equal to that value. Shade the side of dot that fits the solution

Examples: Solve and Graph the Solution

2) 6 + 5(2 – x) ≤ 41

3. 5(2x– 7) > 2x + 9

No Solution or All Real Numbers When your x’s subtract out (cancel out) there are 2 possibilities for the solution: 1. If you are left with a TRUE statement (ex: 0 < 2) then the solution is ALL REAL NUMBERS 2. If you are left with a FALSE statement (ex: 0 > 8) then there is NO SOLUTION

Example: 2x – 3 > 2(x – 5)

Example: 7x + 6 < 7(x – 4)

A pair of inequalities joined by the word And or Or Compound Inequality A pair of inequalities joined by the word And or Or To solve an AND inequality, you have to find all the values that make BOTH inequalities true To solve an OR inequality, you have to find all the values that make At Least 1 inequality true

“OR” Inequalities 1. 3x + 9 < -3 or -2x + 1 ≤ 5

2. 8x > -32 or -6x ≤ 48

Writing the Solution to an “AND” Inequality  The solution to an “and” inequality can be written as a single statement. The solution x > -6 and x ≤ 3 Can be written as -6 < x ≤ 3

“AND” Inequalities 1. -2x < -x – 6 and x – 7 < 3

2. -6 < 2x – 4 ≤ 12

3. 4 ≤ 1 – 3x ≤ 7

On Your Own: Solve each and graph on a number line. 5x – 12 > -37 -12 ≤ x + 6 < -3

Homework Worksheet 4.1