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Find the Solution Region for the System of Linear Inequalities $y \\geq -1/5x + 3$ and $y \\leq 3/4x - 5$
Mathematics
Grade 9 (Junior High School)
Question Content
To complete the graph Jeny must shade the region which represents the solution set to the given system of inequalities. What region of the graph needs to be shaded? \n$\\begin{cases} y \\geq -\\frac{1}{5}x + 3 \\\\ y \\leq \\frac{3}{4}x - 5 \\end{cases}$\nThe graph has 4 labeled regions (1, 2, 3, 4). Answer with 1, 2, 3, or 4.
Correct Answer
4
Detailed Solution Steps
1
Step 1: Analyze the first inequality $y \\geq -\\frac{1}{5}x + 3$. This means we need to shade the region **above or on** the solid line $y = -\\frac{1}{5}x + 3$ (the black line on the graph).
2
Step 2: Analyze the second inequality $y \\leq \\frac{3}{4}x - 5$. This means we need to shade the region **below or on** the solid line $y = \\frac{3}{4}x - 5$ (the blue line on the graph).
3
Step 3: Identify the overlapping region that satisfies both inequalities. Region 4 is the only area that is above the black line and below the blue line, so it is the solution set.
Knowledge Points Involved
1
Graphing Linear Inequalities in Two Variables
For inequalities of the form $y \\geq mx + b$ or $y \\leq mx + b$, solid lines are used to represent the boundary (since the inequality includes equality). For $y \\geq mx + b$, shade above the line; for $y \\leq mx + b$, shade below the line. This rule applies when the inequality is solved for $y$.
2
Solution Set of a System of Linear Inequalities
The solution set of a system of linear inequalities is the intersection (overlap) of the solution sets of each individual inequality. It represents all $(x,y)$ pairs that satisfy every inequality in the system at the same time.
3
Boundary Lines for Linear Inequalities
Solid boundary lines are used when the inequality symbol is $\\geq$ or $\\leq$ (including the line itself in the solution), while dashed lines are used for $>$ or $<$ (excluding the line from the solution).
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