Graphing inequalities on a number line is a fundamental skill in algebra. This worksheet will guide you through the process, explaining the concepts and providing practice problems to solidify your understanding. Mastering this skill is crucial for understanding more complex algebraic concepts.
Understanding Inequalities
Before we delve into graphing, let's review the basics of inequalities. Inequalities compare two expressions, showing that one is greater than, less than, greater than or equal to, or less than or equal to the other. We use the following symbols:
- > Greater than
- < Less than
- ≥ Greater than or equal to
- ≤ Less than or equal to
Graphing Inequalities: The Basics
When graphing inequalities on a number line, we represent the solution set – all the numbers that satisfy the inequality. Here's how it works:
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Open Circle (o): Used for inequalities with > or < (strict inequalities). This indicates that the endpoint is not included in the solution.
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Closed Circle (•): Used for inequalities with ≥ or ≤ (inclusive inequalities). This indicates that the endpoint is included in the solution.
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Shading: The direction of shading indicates the range of solutions. Shade to the right for greater than (>) or greater than or equal to (≥) inequalities, and shade to the left for less than (<) or less than or equal to (≤) inequalities.
Example: Graphing x > 2
Let's graph the inequality x > 2.
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Locate the endpoint: Find 2 on the number line.
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Choose the correct circle: Since it's a "greater than" inequality, we use an open circle (o) at 2.
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Shade the appropriate direction: Because x is greater than 2, we shade to the right of the open circle.
[Insert image here: A number line with an open circle at 2 and shading to the right.]
Example: Graphing y ≤ -1
Now let's graph y ≤ -1.
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Locate the endpoint: Find -1 on the number line.
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Choose the correct circle: Since it's a "less than or equal to" inequality, we use a closed circle (•) at -1.
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Shade the appropriate direction: Because y is less than or equal to -1, we shade to the left of the closed circle.
[Insert image here: A number line with a closed circle at -1 and shading to the left.]
Practice Problems
Now it's your turn! Graph the following inequalities on a number line:
- x < 5
- y ≥ -3
- z > 0
- w ≤ 4
- t > -2
[Insert space here for students to draw number lines and graphs]
Compound Inequalities
Compound inequalities involve two inequality symbols. For example, -2 < x ≤ 5 means that x is greater than -2 and less than or equal to 5. When graphing these, you'll shade the region between the two endpoints. Remember to use the appropriate circle (open or closed) for each endpoint based on the inequality symbol.
Example: Graphing -2 < x ≤ 5
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Locate the endpoints: Find -2 and 5 on the number line.
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Choose the correct circles: Use an open circle (o) at -2 (because it's just greater than) and a closed circle (•) at 5 (because it's less than or equal to).
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Shade the appropriate region: Shade the region between -2 and 5.
[Insert image here: A number line with an open circle at -2, a closed circle at 5, and shading between the two.]
More Practice Problems (Compound Inequalities)
Graph the following compound inequalities on a number line:
- -1 ≤ x < 3
- 0 < y ≤ 4
- -5 < z < 2
[Insert space here for students to draw number lines and graphs]
Troubleshooting Common Mistakes
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Confusing open and closed circles: Remember, open circles are for strict inequalities (> or <), and closed circles are for inclusive inequalities (≥ or ≤).
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Shading in the wrong direction: Always double-check the inequality symbol to ensure you're shading the correct region.
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Misinterpreting compound inequalities: Pay close attention to both inequality symbols and shade the appropriate region between the endpoints.
This worksheet provides a solid foundation for graphing inequalities on a number line. Consistent practice is key to mastering this skill. Remember to always carefully consider the inequality symbols and the meaning of open versus closed circles when creating your graphs.
