Unit 3, Lesson 15
Efficiently Solving Inequalities
Accelerated 7
15.1
Warm-up
Here is an inequality:
- Without testing any values, predict what the solutions to this inequality will look like on the number line.
- Where will the starting (or boundary) point be?
- Will the circle at that point be filled or unfilled?
- Will the shading go to the right or to the left of that point?
- Test each value to see whether it is a solution to the inequality .
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3
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-3
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4
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-4
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4.001
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-4.001
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Graph all possible solutions to the inequality on the number line:
15.2
Activity
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Let's investigate the inequality .
-4 -3 -2 -1 0 1 2 3 4 -7 -6 -4 -1 1 - Complete the table.
- For which values of is it true that ?
- For which values of is it true that ?
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Graph the solutions to on the number line:
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Here is an inequality: .
- Predict which values of will make the inequality true.
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Complete the table. Does it match your prediction?
-4 -3 -2 -1 0 1 2 3 4 -
Graph the solutions to on the number line:
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Here is an inequality: .
- Predict which values of will make the inequality true.
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Complete the table. Does it match your prediction?
-4 -3 -2 -1 0 1 2 3 4 -
Graph the solutions to on the number line:
- How are the solutions to different from the solutions to ?
15.3
Activity
- Let’s investigate .
- Solve .
- Is true when is 0? What about when is 7? What about when is -7?
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Graph the solutions to on the number line.
- Let’s investigate .
- Solve .
- Is true when is 0?
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Graph the solutions to on the number line.
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Solve the inequality , and graph the solutions on the number line.
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Solve the inequality , and graph the solutions on the number line.
Student Lesson Summary
Here is an inequality: . The solution set for this inequality is all the values that can be used in place of to make the inequality true. Each solution is one value that makes the inequality true.
In order to solve this inequality, we can first solve the related equation to get the solution . That means 2 is the boundary between values of that make the inequality true and values that make the inequality false.
To solve the inequality, we can check numbers greater than 2 and less than 2 and see which ones make the inequality true.
Let’s check a number that is greater than 2: . Replacing with 5 in the inequality, we get or just . This is true, so is a solution. This means that all values greater than 2 make the inequality true. We can represent the solutions as and also represent the solutions on a number line:
Notice that 2 itself is not a solution because it's the value of that makes equal to 18, and so it does not make true.
For confirmation that we found the correct solution, we can also test a value that is less than 2. If we test , we get or just . This is false, so and all values of that are less than 2 are not solutions.
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