which graph represents the system of linear inequalities?\n$2x + y < 1$\n$y \\geq \\frac{1}{2}x + 2$
which graph represents the system of linear inequalities?\n$2x + y < 1$\n$y \\geq \\frac{1}{2}x + 2$
Answer
Explanation:
Step1: Analyze (2x + y < 1)
Rewrite as (y < -2x + 1). The boundary line (y=-2x + 1) has a slope of (-2) and y - intercept (1). Since the inequality is (<), the line is dashed, and we shade below it.
Step2: Analyze (y\geq\frac{1}{2}x + 2)
The boundary line (y = \frac{1}{2}x+2) has a slope of (\frac{1}{2}) and y - intercept (2). Since the inequality is (\geq), the line is solid, and we shade above it.
Step3: Find the overlapping region
The solution is the region that is below the dashed line (y=-2x + 1) and above the solid line (y=\frac{1}{2}x + 2). We would check the graphs (not shown here) to see which one has these characteristics: dashed line with slope - 2, solid line with slope (\frac{1}{2}), and the correct shaded region.
(Note: Since the graphs are not provided in the text, but if we assume there are options, we would pick the graph with a dashed line (y = - 2x+1) (slope - 2, y - intercept 1) and a solid line (y=\frac{1}{2}x + 2) (slope (\frac{1}{2}), y - intercept 2), with shading below the dashed line and above the solid line.)
Answer:
(The answer would be the graph that has a dashed line (y=-2x + 1) (slope - 2, y - intercept 1) and a solid line (y=\frac{1}{2}x + 2) (slope (\frac{1}{2}), y - intercept 2), with shading below the dashed line and above the solid line. If there are options labeled, for example, if option A has these features, the answer would be "A. [Description of the correct graph]")