To graph an absolute value function on a number line, understand that the absolute value represents the distance of a number from zero. While graphing functions like y = |x| is typically done on a coordinate plane, representing solutions to simple absolute value equations or inequalities can be done on a number line. Here's how:
Graphing Absolute Value Equations on a Number Line
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Solve the Absolute Value Equation: Isolate the absolute value expression and solve for the variable. Remember that absolute value equations often have two solutions because both a positive and negative value inside the absolute value bars can result in the same positive output.
- Example: |x| = 3. This means x = 3 or x = x = -3
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Represent Solutions on the Number Line: Draw a number line and mark the solutions you found.
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For each solution, draw a closed circle (or a filled-in dot) on the number line at the corresponding value. A closed circle indicates that the value is included in the solution.
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Example: For |x| = 3, place closed circles at 3 and -3 on the number line.
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Graphing Absolute Value Inequalities on a Number Line
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Solve the Absolute Value Inequality: Isolate the absolute value expression and solve the inequality. Absolute value inequalities require careful consideration of two cases (positive and negative).
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Example: |x| < 3. This means -3 < x < 3
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Example: |x| > 3. This means x < -3 or x > 3
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Represent Solutions on the Number Line:
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Less Than (< or ≤):
- For inequalities like |x| < 3, you'll have solutions between two values. Draw an open circle at -3 and 3 (open circles because the inequality is strict - not including -3 and 3) and shade the region between the circles. For |x| ≤ 3, use closed circles instead.
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Greater Than (> or ≥):
- For inequalities like |x| > 3, you'll have solutions outside two values. Draw an open circle at -3 and 3 (open circles because the inequality is strict - not including -3 and 3) and shade the regions to the left of -3 and to the right of 3. For |x| ≥ 3, use closed circles instead.
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Key Considerations
- Open vs. Closed Circles: Use open circles (o) when the value is not included in the solution (for < and >). Use closed circles (•) when the value is included (for ≤ and ≥).
- Shading: Shading indicates the range of numbers that satisfy the inequality.
Example
Let’s graph |x - 1| ≤ 2 on a number line.
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Solve: -2 ≤ x - 1 ≤ 2. Adding 1 to all parts gives -1 ≤ x ≤ 3.
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Graph: Draw a number line. Place closed circles at -1 and 3. Shade the region between -1 and 3, indicating all values of x that satisfy the inequality.
Graphing Absolute Value Functions
According to the provided reference, to graph absolute value functions, plot two lines for the positive and negative cases that meet at the expression's zero. The graph is v-shaped. However, this typically refers to graphing on a coordinate plane, not just on a number line. While you can identify key points (like where the "V" occurs) and represent those specific values on a number line, the number line alone won't show the entire function's V-shape. Instead, it will show the solution to the equation set to a particular value.
