Learn Graphing Inequalities - MR. JONES · Solve linear inequalities by using addition. Solve linear inequalities by using subtraction. Lesson 6-1 • Solving One-Step Inequalities

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Solving One-Step Inequalities

Lesson 6-1

Today’s Vocabulary inequalityset-builder notation

Explore Graphing Inequalities

Online Activity Use graphing technology to complete the Explore.

Think About It!

Compare and contrast the graphs of x ≥ 6 and x > 6.

Think About It!

How do you know which inequality symbol to use by looking at a graph?

Go Online You can complete an Extra Example online.

INQUIRY How can you graph the solution set of an inequality of the form x < a or x > a for some number a?

Learn Graphing InequalitiesAn inequality is an open sentence that contains the symbol <, >, ≤, or ≥.

Example 1 Graph InequalitiesGraph the solution set of y ≤ 4.

-4 -3 -2 -1 0 1 2 3 4 5 6

The dot at 4 shows that 4 a solution. The heavy arrow pointing to the shows that the solution includes all numbers 4.

Check Graph the solution set of y > 1 _ 2 .

-2 -1 0 1 2 3

Example 2 Write Inequalities from a GraphWrite an inequality that represents the graph.

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3The endpoint is shown with a circle at 1.4, so 1.4 is in the solution. The inequality must be < or >.

The arrow points to values than 1.4. The graph represents the solution of .

Check Write an inequality that represents the graph.

-10 -8 -6 -4 -2 0 2 4 6 8 10

Today’s Goals● Graph the solutions of

an inequality.● Solve linear inequalities

by using addition.● Solve linear inequalities

by using subtraction.

Lesson 6-1 • Solving One-Step Inequalities 341

Sample answer: Both graphs have the endpoint at 6. The graph of x ≥ 6 has a dot at 6 with an arrow pointing to the right. The graph of x > 6 has a circle at 6 with an arrow pointing to the right.

Sample answer: Check to see whether the endpoint is included in the solution. Then use the graph to see whether the solution set includes values greater than or less than the endpoint.

left less than

not included

a < 1.4

x ≥ −10

less

is

THIS MATERIAL IS PROVIDED FOR INDIVIDUAL EDUCATIONAL PURPOSES ONLY AND MAY NOT BE DOWNLOADED, REPRODUCED, OR FURTHER DISTRIBUTED.

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Your Notes


Explore Properties of Inequalities

Online Activity Use graphing technology to complete the Explore.

Talk About It!

How many solutions of the inequality are there? Justify your argument.

Study Tip

Set-Builder Notation {x | x < 25} is read the set of all numbers x such that x is less than 25.

INQUIRY Do the properties of equality hold true for inequalities?

Learn Solving Inequalities by Using Addition and SubtractionAddition and subtraction can be used to solve inequalities.

Key Concept • Addition Property of InequalitiesWords If the same number is added to each side of a true

inequality, the resulting inequality is also true.Symbols For any real numbers a, b, and c, the following are true.

If a > b, then a + c > b + c.If a < b then a + c < b + c.

Key Concept • Subtraction Property of InequalitiesWords If the same number is subtracted from each side of a true

inequality, the resulting inequality is also true.Symbols For any real numbers a, b, and c, the following are true.

If a > b, then a − c > b − c.If a < b then a − c < b − c.

When solving inequalities, you can write the solution set in a more concise way using set-builder notation. For example, {x | x ≥ −4} represents the set of all numbers x such that x is greater than or equal to −4.

Example 3 Solve Inequalities by AddingSolve x − 10 < 15.

x − 10 < 15 Original inequality

x – 10 + 10 < 15 + 10 Add 10 to each side to isolate x.

x < Simplify.

The solution set is { }.

Check Solve −9 + b ≤ 16.

342 Module 6 • Linear Inequalities

25x | x < 25

{b | b ≤ 25}

Sample answer: Infinitely many; there are an infinite number of values less than 25.


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Study Tip

Writing Inequalities Simplifying the inequality so that the variable is on the left side, as in y ≤ 3, prepares you to write the solution set in set-builder notation and graph the inequality on a number line.

Think About It!

How can you check the solution of the inequality?

Example 4 Solve Inequalities by SubtractingSolve x + 24 ≥ 61.

x + 24 ≥ 61 Original inequality

x + 24 – 24 ≥ 61 – 24 Subtract 24 from each side.

x ≥ Simplify.


Check Select the solution set for 88 < x + 13.

Example 5 Add or Subtract to Solve Inequalities with Variables on Each SideSolve 9y + 3 ≥ 10y.

9y + 3 ≥ 10y Original inequality

9y – 9y + 3 ≥ 10y – 9y Subtract 9y from each side.

≥ y Simplify.

Since 3 ≥ y is the same as y ≤ 3, { }.

CheckSelect the solution set for 7x + 6 < 8x.

Example 6 Use an Inequality to Solve a ProblemDATA USAGE Hassan’s wireless contract allows him to use at most 5 gigabytes (GB) of data per month. At this point, Hassan has used 3.7 GB of data. How many gigabytes of data can Hassan use during the rest of the month without exceeding the maximum allowance?

Complete the table to write an inequality to represent how many gigabytes of data Hassan can use. Then solve the inequality.

Words Hassan can use 5 GB of data.

Variables Let g = the that Hassan has left to use

Inequality

(continued on the next page)


37x | x ≥ 37

y | y ≤ 33

53.7 + g ≤

number of gigabytes

{x | x > 75}

{x | x > 6}

Sample answer: Substitute 37 into the original inequality. Then test values greater than or less than 37 to confirm the direction of the inequality symbol.

at most


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Use a Source

Research data plans for wireless carriers in your area. Write and solve your own inequality to represent the amount of data remaining if you have already used 5.2 GB.

Study Tip

Inequalities Verbal problems containing phrases like greater than and less than can be solved by using inequalities. Some other phrases that include inequalities are:

< less than; fewer than> greater than; more than≤ less than or equal to; at most; no more than≥ greater than or equal to; at least; no less than

Think About It!

If a, b, and c are positive real numbers, what must be true if ac is greater than or equal to bc?

What must happen to an inequality symbol when you divide each side by a negative number if the inequality is to remain true?

3.7 + g ≤ 5 Original inequality

3.7 - 3.7 + g ≤ 5 - 3.7 Subtract 3.7 from each side.

g ≤ Simplify.


Hassan can use up to GB of data without exceeding his maximum allowance. Notice that negative numbers are solutions to the inequality, but they are solutions to the problem because Hassan cannot use a negative amount of data.

Learn Solving Inequalities by Using Multiplication and DivisionIf you multiply or divide each side of an inequality by a positive number, then the inequality remains true.

If you multiply or divide each side of an inequality by a negative number, the inequality symbol changes direction.

Key Concepts • Multiplication Property of InequalitiesWords If each side of a true

inequality is multiplied by a positive number, the resulting inequality is also true.

If each side of a true inequality is multiplied by a negative number, the direction of the inequality sign must be reversed to make the resulting inequality also true.

Symbols For any real numbers a and b and any positive real number c:

If a > b, then ac > bc.

If a < b then ac < bc

For any real numbers a and b and any positive real number c:

If a > b, then ac < bc.

If a < b then ac > bc

Key Concepts • Division Property of InequalitiesWords If each side of a true

inequality is divided by a positive number, the resulting inequality is also true.

If each side of a true inequality is divided by a negative number, the direction of the inequality sign must be reversed to make the resulting inequality also true.

Symbols For any real numbers a and b and any positive real number c:

If a > b, then a __ c > b __ c

If a < b then a __ c < b __ c

For any real numbers a and b and any negative real number c:

If a > b, then a __ c < b __ c .

If a < b then a __ c > b __ c .

These properties also hold true for inequalities involving ≤ and ≥.


Sample answer: a must be greater than or equal to b.

Sample answer: The inequality symbol must be reversed.

Sample answer: 5.2 + g ≤ 16; g ≤ 10.8 gigabytes

1.3

1.3

not viable

g | g ≤ 1.3


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Apply Example 7 Write and Solve an InequalityBOOKS Alisa has read 1 _ 4 of a novel. If she has read at least 112 pages, how many pages are there in the novel?

1. What is the task? Describe the task in your own words. Then list any questions that you may have. How can you find answers to your questions?

2. How will you approach the task? What have you learned that you can use to help you complete the task?

3. What is your solution? Estimate the number of pages in the novel.

Write an inequality to represent this situation. Let n = the number of pages in the novel.

There are at least pages in the novel.4. How can you know that your solution is reasonable?

Write About It! Write an argument that can be used to defend your solution.

Check ELECTRIC CAR For every hour x that Eva’s electric car charges, she can drive the car 7.5 miles. Eva needs to drive at least 60 miles tomorrow.

Part A What inequality represents the situation in terms of x hours?

Part B What is the least amount of time that Eva will need to charge her car? hours



8

7.5x ≥ 60

Sample answer: I know the number of pages read and the fraction of the novel read. I need to find out how many pages are in the novel.

Sample answer: I will use estimation first. Then I will write an inequality to represent the situation and solve it.

Sample answer: Use multiplication; 448 ( 1 _ 4 ) = 112, so 448 is reasonable. Also, 448 > 400 which makes sense with our estimate of more than 400 pages.

400

448

1 __ 4 · n ≥ 1124 ( 1 _ 4 ) n ≥ 4(112)

n ≥ 448


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Study Tip

Multiplicative Inverses The multiplicative inverse, or reciprocal, of a number can be used to undo multiplication. Multiplying − 2 __ 5 x by the reciprocal − 5 __ 2 in the example at the right is the same as dividing by − 2 __ 5 , but is easier to compute mentally.

Think About It!

Why was the inequality symbol reversed?

Study Tip

Negatives A negative sign in an inequality does not necessarily mean that the direction of the inequality symbol should change. For example, when solving x _ 3 ≥ −9, do not change the direction of the inequality symbol.

Think About It!

Why is the solution of −13z ≥ 117 shaded to the left when the original inequality symbol is greater than or equal to?

B

Example 8 Solve an Inequality by MultiplyingSolve - 2 __ 5 x ≤ 11. Graph the solution set on a number line.

- 2 __ 5 x ≤ 11 Original inequality

- 2 __ 5 x ≤ 11

x Simplify.


-30 -29 -28 -27 -26 -25

Example 9 Solve an Inequality by DividingSolve 20x < 4. Graph the solution set on a number line.

20x < 4 Original inequality

20x ___ 20 < 4 __ 20 Divide each side by 20.

x < Simplify.


4-5 -2-3 -1 0 1 32-4 5

Check Solve 7x + 6 < 8x.

Example 10 Solve an Inequality with a Negative CoefficientSolve -13z ≥ 117. Graph the solution set on a number line.

-13z ≥ 117 Original inequality

- 13z ___ -13 ≥ 117 ___ -13 Divide each side by -13.

z ≤ Simplify.


-16 -14 -12 -10 -8 -6 -4 -2 0 2 4

CheckSelect the solution set for -13x > −169.

A. {x | x > 13} B. {x | x < 13} C. {x | x > −13} D. {x | x <−13}

Multiply each side by - 2 __ 5 . Reverse the inequality symbol.



Sample answer: When you multiply or divide an inequality by a negative number, you need to reverse the direction of the inequality symbol to make the resulting inequality true.

Sample answer: When you multiply or divide by a negative number, you reverse the direction of the inequality symbol. The new inequality symbol is less than or equal to.

≥ -27.5

(- 5 __ 2 ) (- 5 __ 2 )

1 __ 5

x | x ≥ −27.5

{x | x > 6}

z | z ≤ -9

-9

x | x < 1 __ 5


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Learn Graphing Inequalities - MR. JONES · Solve linear inequalities by using addition. Solve linear inequalities by using subtraction. Lesson 6-1 • Solving One-Step Inequalities