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Identify the Linear Inequality Represented by a Graph
Mathematics
Grade 9 (Junior High School)
Question Content
Which of the inequalities is represented by the graph? The graph shows a solid line with a negative slope passing through points (0,2) and (2,0), with the region below the line shaded. The options are: y≥2x-1, y<x+2, y≤-x+2, y<-x+2
Correct Answer
y≤-x+2
Detailed Solution Steps
1
Step 1: Determine the equation of the boundary line. Identify two points on the line: (0, 2) (y-intercept, so b=2) and (2, 0). Calculate the slope m using the formula m=(y₂-y₁)/(x₂-x₁) = (0-2)/(2-0) = -1. Using slope-intercept form y=mx+b, the line equation is y = -x + 2.
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Step 2: Determine if the line is solid or dashed. The line in the graph is solid, which means the inequality includes equality (≤ or ≥), so we can eliminate the options with strict inequalities: y<x+2 and y<-x+2.
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Step 3: Test a point in the shaded region to confirm the inequality direction. Choose the origin (0,0), which is in the shaded area. Substitute into the remaining options: For y≥2x-1: 0≥2(0)-1 → 0≥-1, which is true, but this line has a positive slope that does not match the graph. For y≤-x+2: 0≤-0+2 → 0≤2, which is true, and this matches the line equation and shaded region.
Knowledge Points Involved
1
Slope-Intercept Form of a Line
The slope-intercept form is y = mx + b, where m is the slope (rate of change of y with respect to x) and b is the y-intercept (the point where the line crosses the y-axis). It is used to quickly write the equation of a line when the slope and y-intercept are known, and is essential for graphing linear functions and inequalities.
2
Linear Inequality Graph Boundary Rules
When graphing linear inequalities, a solid boundary line is used if the inequality includes equality (≤ or ≥), meaning points on the line are part of the solution set. A dashed boundary line is used for strict inequalities (< or >), meaning points on the line are not part of the solution set.
3
Testing Points for Inequality Solution Regions
To confirm which side of the boundary line is the solution region, pick a test point not on the line (usually the origin (0,0) if it is not on the line) and substitute its coordinates into the inequality. If the resulting statement is true, the side containing the test point is the solution region; if false, the opposite side is the solution.
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