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 examples to solidify your understanding. Mastering this skill is crucial for solving and representing a wide range of algebraic problems.
Understanding Inequalities
Before we dive into graphing, let's review the symbols used to represent inequalities:
- < less than
- > greater than
- ≤ less than or equal to
- ≥ greater than or equal to
- ≠ not equal to
These symbols describe the relationship between two expressions. For instance, x < 5 means that the variable x can represent any number less than 5, but not 5 itself. y ≥ -2 means y can be -2 or any number greater than -2.
Graphing Inequalities on a Number Line
A number line is a visual representation of numbers. To graph an inequality, we mark the relevant numbers on the line and use shading to represent the solution set.
Steps to Graphing an Inequality:
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Identify the inequality symbol: Determine whether the inequality is less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥).
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Locate the critical value: The critical value is the number being compared to the variable. For example, in
x > 3, the critical value is 3. -
Mark the critical value on the number line: Draw a number line and place a dot at the critical value.
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Determine the type of circle:
- If the inequality is less than (<) or greater than (>), use an open circle (◦) to indicate that the critical value is not included in the solution.
- If the inequality is less than or equal to (≤) or greater than or equal to (≥), use a closed circle (•) to indicate that the critical value is included in the solution.
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Shade the appropriate region:
- For inequalities with "<" or "≤", shade the region to the left of the critical value.
- For inequalities with ">" or "≥", shade the region to the right of the critical value.
Examples
Let's work through some examples to illustrate the process:
Example 1: x > 2
- Symbol: > (greater than)
- Critical Value: 2
- Circle: Open circle (◦) because 2 is not included.
- Shading: Shade to the right of 2.
[Visual representation of a number line with an open circle at 2 and shading to the right]
Example 2: y ≤ -1
- Symbol: ≤ (less than or equal to)
- Critical Value: -1
- Circle: Closed circle (•) because -1 is included.
- Shading: Shade to the left of -1.
[Visual representation of a number line with a closed circle at -1 and shading to the left]
Example 3: z ≠ 0
- Symbol: ≠ (not equal to)
- Critical Value: 0
- Shading: Shade everything except 0. This will be represented by open circles at 0 and shading to both the left and right.
[Visual representation of a number line with open circles at 0 and shading to both left and right]
Compound Inequalities
Compound inequalities involve two inequality symbols. For example: -2 ≤ x < 5 This means x is greater than or equal to -2 AND less than 5. When graphing this, you would shade the region between -2 and 5, including -2 but excluding 5.
[Visual representation of a number line with a closed circle at -2, an open circle at 5, and shading between them]
Practice Problems
Now it's your turn! Try graphing these inequalities on a number line:
- x < -3
- y ≥ 4
- z ≤ 0
- -1 < x ≤ 2
- x ≠ 5
This worksheet provides a solid foundation for understanding and graphing inequalities. Remember to practice regularly to master this essential algebraic skill. Further practice problems can be found in various algebra textbooks and online resources.
