Math Problem Statement
The query submitted by the user involving solving a system of inequalities.
Solution
The image you provided outlines a step-by-step process for solving a system of inequalities. Let's go through the content in the image to help you understand and complete it:
Steps to Solve the System of Inequalities:
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Put the system of inequalities in general form:
- Given:
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Write one inequality in slope-intercept form (if applicable):
- For , solve for :
- This form helps with graphing by showing the slope and y-intercept.
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Graph the inequalities on a coordinate plane:
- For the inequality :
- Plot the line as a solid line because the inequality includes "equal to" ().
- Shade below the line since is less than or equal to the line.
- For the inequality :
- Plot the vertical line as a solid line.
- Shade to the right of the line since is greater than or equal to 2.
- For the inequality :
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Find the critical values in the shaded region:
- Identify where the shaded regions overlap. This overlapping area is the solution to the system of inequalities.
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Label the area of interest on the graph:
- Clearly indicate the feasible region that satisfies both inequalities.
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Check a point in the solution region:
- Verify that the point satisfies both inequalities.
Questions Related to This Topic:
- How can you determine if a line should be dashed or solid when graphing inequalities?
- What does the slope-intercept form of a line reveal about the line?
- How can you verify the solution region of a system of inequalities?
- What is the significance of shading in graphing inequalities?
- How do you handle systems of inequalities with more than two inequalities?
Tip:
Always double-check your shaded regions when graphing inequalities to ensure that you have the correct overlap for your solution set.
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Math Problem Analysis
Mathematical Concepts
Algebra
Inequalities
Graphing
Formulas
Slope-intercept form y = mx + b
Theorems
Linear inequality theorem
Suitable Grade Level
Grades 8-10