Solving and Graphing Inequalities in One Variable
Scenario: You are studying Earth’s atmosphere in science class. You learn that the atmosphere is made of four layers: the troposphere, the stratosphere, the mesosphere, and the thermosphere. Beyond the thermosphere is the exosphere, which extends until the atmosphere mixes with gases in space.
“The Layers Above”
In your science class, you are given the following information about these layers of the atmosphere.
Layer Location Troposphere Starts at Earth’s surface and extends to 9 miles above Earth’s surface Stratosphere Starts just above troposphere and extends to 31 miles above Earth’s surface Mesosphere Starts just above stratosphere and extends to 53 miles above Earth’s surface Thermosphere Starts just above mesosphere and extends to 372 miles above Earth’s surface Exosphere Starts at top of thermosphere and extends upward
A. How many miles above the surface of Earth does the troposphere start? Use a complete sentence in your answer.
B. How many miles above the surface of Earth does the troposphere end? Use a complete sentence in your answer.
C. Does the stratosphere start at 9 miles above the surface of Earth? Use complete sentences to explain.
D. Does the exosphere start at 372 miles above the surface of Earth? Use complete sentences to explain.
1. Complete each statement below with “greater than,” “less than,” “greater than or equal to,” or “less than or equal to.”
The height of the troposphere above Earth’s surface is ______0 miles and is ______9 miles.
The height of the stratosphere above Earth’s surface is ______9 miles and is ______31 miles.
The height of the mesosphere above Earth’s surface is ______31 miles and is ______53 miles.
The height of the thermosphere above Earth’s surface is ______53 miles and is ______372 miles.
The height of the exosphere above Earth’s surface is ______372 miles.
2. You can use inequalities to represent the layers of Earth’s atmosphere. An inequality is a statement that is formed by placing an inequality symbol between two expressions. Inequality symbols are <, >, ≤, and ≥.
The symbol < means “less than.” The symbol > means “greater than.” The symbol ≤ means “less than or equal to.” The symbol ≥ means “greater than or equal to.” Adapted for educational use only from Carnegie Learning. Pittsburgh: Carnegie Learning, Inc., 2008. Print. If you let m represent the number of miles above Earth’s surface, you can write an inequality to represent each layer’s position above Earth’s surface. Complete each inequality using an inequality symbol.
Troposphere: m ____ 0 and m ____ 9 Stratosphere: m ____ 9 and m ____ 31 Mesosphere: m ____ 31 and m ____ 53 Thermosphere: m ____ 53 and m ____ 372 Exosphere: m ____ 372
3. All but the last inequality that you wrote in Question 2 are compound inequalities. A compound inequality is formed when two inequalities are connected by the word and or the word or. If you let n represent any number, then you can write the two examples of compound inequalities below. 1 n 5 n 3 or n > 6 The first inequality is read as “all numbers great than 1 and less than or equal to 5.” The second inequality is read as “all numbers less than or equal to –3 or greater than 6.” Notice that 1 n 5 is a compact form of the inequality 1 n and n 5 . The inequality can also be written as n 1 and n 5 .
Write your compound inequalities from Question 2 by using the compact form.
Troposphere: ______
Stratosphere: ______
Mesosphere: ______
Thermosphere: ______
Exosphere: ______
4. You can use a number line to represent your inequalities. The graph of an inequality in one variable is the set of all points that make the inequality true. For instance, the graph of the inequality x 4 consists of the number 4 and all numbers smaller than 4, as shown below.
The endpoint 4 of the graph x 4 is a solid dot because 4 is a solution of the inequality. If the endpoint of the graph is not a solution, draw an open dot for the endpoint.
Write an inequality for each graph shown.
______
______
______
______
Adapted for educational use only from Carnegie Learning. Pittsburgh: Carnegie Learning, Inc., 2008. Print. 5. Graph the inequalities that you wrote in Question 3.
Troposphere:
Stratosphere:
Mesosphere:
Thermosphere:
Exosphere:
“Aircraft in the Atmosphere”
Different aircraft can travel in different layers of the atmosphere. Aircraft have what is called a service ceiling, or the highest altitude at which they can fly without stalling. The service ceilings of different aircraft are given in the table below.
Aircraft Blackhawk Boeing 747 Concorde X-15A-2 X-33 Aircraft type Helicopter Commercial Commercial Research Research plane plane plane plane Approximate service ceiling (miles) 4 6 11 67 47
A. An X-33 aircraft is currently at an altitude of 15 miles. Can the X-33 climb 20 miles and stay under the service ceiling?
Can the X-33 climb 26 miles and stay under the service ceiling?
Can the X-33 climb 32 miles and stay under the service ceiling?
Can the X-33 climb 35 miles and stay under the service ceiling?
B. Write and graph an inequality that describes the possible number of miles that the X-33 can climb from 15 miles and still be at or below its service ceiling. Use m to represent the number of miles that the X-33 climbs.
1. You were actually writing and solving an inequality. To solve an inequality means to find the values of the variable that make the inequality true. The steps to solving an inequality are similar to the steps of solving an equation. That is, you want to get the variable by itself on one side of the inequality symbol by using the operations of addition, subtraction, multiplication, and division.
What steps would you take to solve the equation 2x 1 5? Use complete sentences in your answer.
Adapted for educational use only from Carnegie Learning. Pittsburgh: Carnegie Learning, Inc., 2008. Print. You can use the same exact steps to solve the inequality 2x 1 5 . Solve the inequality and then graph the solution. Show all your work.
2. You can check your solution to an inequality by choosing a value that is in the solution set and substituting it into the original inequality. Choose a value in your solution of the inequality 2x 1 5 and show that it is a solution.
3. What is the solution to the equation 2x 1 5? How do you think you should graph the solution? Explain your reasoning. Use complete sentences in your answer.
4. Whenever you multiply or divide each side of an inequality by a negative number, you have to reverse the inequality symbol. For instance, to solve the inequality 2x 14 , divide each side by –2 and reverse the inequality symbol. 2x 14 2x 14 2 2 x 7
Solve each inequality and graph the solution. x 4x > 4 1 3
x 2x 4 2 4
5. Suppose that the Boeing 747 is currently at an altitude of 2 miles. Use m to represent the number of miles that the Boeing 747 climbs. Write an expression that represents the Boeing 747’s altitude if it climbs m miles from its current altitude.
Now write and solve an inequality that describes the possible altitudes to which the Boeing 747 can climb and still be at or below the service ceiling. Show all your work. Use a complete sentence to summarize these altitudes.
6. Suppose that the X-15A-2 is currently at an altitude of 28 miles. Use m to represent the number of miles that the X-15A-2 climbs. Write an expression that represents the X-15A-2’s altitude if it climbs m miles from its current altitude.
Now write and solve an inequality that describes the possible altitudes to which the X-15A-2 can climb and still be at or below the service ceiling. Show all your work. Use a complete sentence to summarize these altitudes.
Adapted for educational use only from Carnegie Learning. Pittsburgh: Carnegie Learning, Inc., 2008. Print.