S1 OBJECTIVES (AIM): Students will: Define the term inequality. Solve inequalities with one variable. Interpret inequalities by graphing. 1

S1 DO NOW: Complete these problems: Explain what is different about questions 3&4. How does this change the answer? An equation may have only one solution, where as an inequality may have multiple solutions.

>, <, ≥ or ≤ Inequalities An inequality is an algebraic statement involving the symbols >, <, ≥ or ≤ For example, x > 3 means ‘x is greater than 3’. x < –6 means ‘x is less than –6’. x ≥ –2 means ‘x is greater than or equal to –2’. x ≤ 10 means ‘x is less than or equal to 10’. For each example ask pupils to give you some values that x could have. Praise more imaginative answers involving fractions or decimals. Tell pupils that we can combine two inequalities as shown whenever the same variable falls between two values. Sometime two inequalities can be combined in a single statement. For example, If x > 3 and x ≤ 14 we can write 3 < x ≤ 14

Representing inequalities on number lines Suppose x > 2. There are infinitely many values that x could have. x could be equal to 3, 7.3, 54 , 18463.431 … 3 11 It would be impossible to write every solution down. We can therefore represent the solution set on a number line as follows: –3 –2 –1 1 2 3 4 5 6 7 A hollow circle, , at 2 means that this number is not included and the arrow at the end of the line means that the solution set extends in the direction shown.

Representing inequalities on number lines Suppose x ≤ 3. Again, there are infinitely many values that x could have. x could be equal to 3, –1.4, –94 , –7452.802 … 8 17 We can represent the solution set on a number line as follows, –3 –2 –1 1 2 3 4 5 6 7 A solid circle, , at 3 means that this number is included and the arrow at the end of the line means that the solution set extends in the direction shown.

Representing inequalities on number lines Suppose –1 ≤ x < 4. Although x is between two values, there are still infinitely many values that x could have. x could be equal to 2, –0.7, –3 , 1.648953 … 16 17 We can represent the solution set on a number line as follows: –3 –2 –1 1 2 3 4 5 6 7 It may not be obvious to pupils that between any two numbers there are infinitely many real numbers. This is due to the fact that a number line is continuous with no gaps. This means that any two values, no matter how close, would always have infinitely many other values between them. A solid circle, , is used at –1 because this value is included and a hollow circle, , is used at 4 because this value is not included. The line represents all the values in between.

Class work Try these!! A. Graph the following examples 1. 2. 3. B. State the solution set for each of the graphs. 1. 2. 3. –1 1 2 3 4 5 6 7 8 9 -6 -5 -4 -3 -2 -1 1 2 3 4 Complete in your notebook –1 1 2 3 4 5 6 7 8 9

Solving linear inequalities Look at the following inequality, x + 3 ≥ 7 What values of x would make this inequality true? Any value of x greater or equal to 4 would solve this inequality. We could have solved this inequality as follows, x + 3 ≥ 7 When we talk about solving an inequality we are talking about finding the values of x that make the inequality true. Start by finding the solution by trial and improvement. For example, if x = 2 we have 2 + 3 ≥ 7. This is untrue and so x cannot be equal to 2. If x = 4 we have 4 + 3 ≥ 7. This is true and so x is a solution. Discuss the fact that any number less then 4 would not satisfy the inequality. Any number greater than 4 would satisfy the inequality and so the solution is x ≥ 4. Point out that when we solve an inequality we line up the inequality sign in the same way as solving an equation. x + 3 – 3 ≥ 7 – 3 subtract 3 from both sides: x ≥ 4 The solution has one letter on one side of the inequality sign and a number on the other.

Closure Complete these problems: Determine the solution to the inequality, and graph. An equation may have only one solution, where as an inequality may have multiple solutions.

homework: Solving equations Mid-year Review – attempt at all questions Due Jan 18, 09