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6.1: Inequalities Using Addition and Subtraction

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Learning Objectives

At the end of this lesson, students will be able to:

  • Write and graph inequalities with one variable on a number line.
  • Solve an inequality using addition.
  • Solve an inequality using subtraction.

Vocabulary

Terms introduced in this lesson:

inequality
interval, interval of values

Teaching Strategies and Tips

Use Examples 1 and 2 to introduce the number line as a way to graph the solution set to an inequality.

  • Point out that the solution sets in Examples 1 and 2 represent all numbers which make the statements true. The solution to a linear equation is a number; the solution to an inequality, an interval of (infinite) numbers.
  • Remind students that an open circle is used for inequalities containing a > or < symbol and a closed circle for inequalities containing a \ge or \le symbol.

Additional Examples:

Graph the following inequalities on the number line.

a. x < -6

b. x \ge 3

c. x > 1

d. x \le 10

Write the inequality that is represented by each graph.

a.

b.

c.

d.

In Example 3, students learn to identify inequalities in sentences. The following chart might be useful for those students having difficulty choosing the correct symbol.

& < && > && \le && \ge\\& \text{less than} && \text{greater than} && \text{at most} && \text{at least}\\& \text{fewer} && \text{more than} && \text{no more than} && \text{no less than}\\& &&&& \text{less than or equal to} && \text{greater than or equal to}

Additional Examples:

Write each statement as an inequality and graph it on the number line.

  • You were told not to spend any more than \$20 at the arcade.
  • Fewer than 200 tickets are available for sale to the musical performance.
  • You must be taller than 40\;\mathrm{inches} to get on this ride.
  • The FDA allows for 30 or more insect fragments per 100\;\mathrm{grams} of peanut butter.

Use Examples 4 and 5 to show students how to isolate variables in inequalities using addition and subtraction.

  • Point out that solving inequalities is analogous to solving equations.
  • The exception occurs when multiplying or dividing by a negative number. See the lesson Inequalities Using Multiplication and Division.

Error Troubleshooting

General Tip: Suggest to students having difficulty with inequalities that the inequality opens to the larger number.

  • a < b is read as “a is less than b” or “b is greater than a,” depending on the perspective.
  • In a statement such as 2 < x, suggest that students take the point of view of the unknown: “x is greater than 2” or “all real numbers greater than 2” instead of “2 is less than x”.

In Example 5d, remind students of mixed number form and that - \frac{3} {4} - 5 = -5 \frac{3} {4}.

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