Students will review the concept of translating inequalities.
1) Write as an inequality, using h as the height: All riders on the roller coaster must be at least 48 inches tall.
2) Write as an inequality, using a as the amount of money: Tommy is not sure how much money he has saved. He has greater than $10, but no more than $23.
The first problem is similar to expressions that students worked on in the previous lesson, Writing Algebraic Inequalities. The second problem adds a deeper level of understanding so I will ask students questions to help them with the translation.
How is problem 2 different from problem 1?
Students may notice that problem 2 refers to a range of values.
Is there a minimum amount of money Tommy has saved? Is there a maximum?
The phrase "greater than" indicates the minimum and "no more than" indicates the maximum.
Can you write this as two inequalities?
Students should be able to write the inequalities: a > 10 and a ≤ 23
Can we combine these into one inequality?
Students may have more difficulty with the compound inequality. 10 < a ≤ 23
Before students begin graphing inequalities, we will explore the meaning of an inequality and the solutions that make an inequality true.
x = 3 has how many solutions?
Students may think that it is a "trick" question, so I will clarify with follow up questions.
Do we need to solve for x? Do we know what x is? Can x be 10 or any other number?
Students should realize that for this equation, x can only be 3.
x > 3 has how many solutions?
Students may realize that for this inequality, x has a lot of solutions. For students who are confused, I will ask a few follow up questions.
Can x be 3.1? 12? 4 billion?
Students should understand that x is any number larger than 3.
For this inequality, x has infinitely many solutions.
I will explain to students that there is a way to show the solution to an inequality. We will formulate steps as we complete an example together.
Example 1 - Graph x >3
Step 1 - Draw a number line.
Step 2 - Draw a circle at the number in the inequality sentence.
What number should we draw a circle at?
Step 3 - If the symbol is < or >, leave the circle open.
If the symbol is ≥ or ≤, color in the circle completely.
Should we close the circle or leave it open?
Step 4 - Shade in the direction of the solution set.
What is a possible value for x?
We will continue with a few more examples. I will ask the same questions of students to help them through the steps of graphing. Examples 4 and 5 may be more difficulty for students, so I will ask additional questions.
Example 2 - Graph x ≤ 1
Example 3 - Graph y < -4
Example 4 - Graph -5 < x ≤ 4
What is different about this inequality?
Students should recognize that this is a compound inequality with a minimum and maximum value.
What is the minimum? What is the maximum? How can we graph these on the number line?
Similar to the steps of the previous examples, students may realize that we can use circles to indicate the minimum and maximum.
What type of circles should we have at each number?
An open circle should be place at -5 and a closed circle at 4.
How can we indicate that the solution to this inequality is between these two numbers and not beyond them?
We will shade only between the values.
Example 5 - Graph -3 ≤ x
What is different about this example?
Students should notice that the number is first and the variable is second.
How would we interpret this inequality?
Students should suggest either "-3 is less than or equal to x" or "x is greater than or equal to -3".
What value of x makes this true?
Students should give answers of -3 or greater. This will help them when determine which direction to shade the number line.
The Independent Practice will assess students' understanding of the lesson. Students will have to create a number line and graph each inequality. To ensure that students understand the inequality and its' solutions, they will have to give a possible value that would be part of the solution set.
Students may have difficulty with 3 - 5 because these needed to be translated first. I will suggest to students that they use their Writing Algebraic Inequalities Chart to help them interpret and translate the key words.
Graph the solution to each inequality. Write one value that could be a possible solution.
1) x < -1
2) 0 < y
3) Carl has at least $5.
4) Tyrone has more than $3.
5) Vanessa has at most $8.
After 10 minutes, we will discuss students' work and answers.
It is important that students are able to interpret inequalities that have already been graphed. For the Graphing Inequalities Lesson Summary, students will answer the following questions:
Is there a minimum value for the inequality?
Is there a maximum value for the inequality?
What is a possible solution for the inequality?
What is the inequality?