Question

an entrepreneur has signboard stands to hang at most 4 banners or 6 signs for a field promotion. each stand can be used. each banner costs $3 and each sign costs $1 per stand. if x is the number of banners and y is the number of signs, which graph represents the system of inequalities for this situation?

Explanation:

Step1: Define Variables and Constraints

Let \( x \) = number of hens, \( y \) = number of roosters.

• At most 4 animals: \( x + y \leq 4 \) (line \( y = -x + 4 \), shaded below, solid line).
• Cost: \( 15x + 5y \leq 15 \) → Simplify: \( 3x + y \leq 3 \) (line \( y = -3x + 3 \), shaded below, solid line).
• Non - negative: \( x \geq 0, y \geq 0 \) (first quadrant).

Step2: Analyze Intercepts and Lines

• For \( x + y = 4 \): Intercepts \( (4,0) \), \( (0,4) \).
• For \( 3x + y = 3 \): Intercepts \( (1,0) \), \( (0,3) \).
• The feasible region is where \( x,y \geq 0 \), below both lines. The line \( x = 0 \) (y - axis) and \( y = 0 \) (x - axis) bound the first quadrant. The correct graph will have these two lines and shading in the overlapping first - quadrant region that satisfies both inequalities. The line \( 3x + y = 3 \) is steeper than \( x + y = 4 \), and the feasible region is near the origin, bounded by \( x\geq0 \), \( y\geq0 \), \( x + y\leq4 \), and \( 3x + y\leq3 \).

Answer:

The graph with the feasible region bounded by \( x\geq0 \), \( y\geq0 \), the line \( y=-x + 4 \) (solid, shaded below), the line \( y=-3x + 3 \) (solid, shaded below), and the axes. Typically, this would be a region in the first quadrant, below both lines, with vertices at \((0,0)\), \((0,3)\), \((1,0)\) (and also considering the intersection of \( x + y = 4 \) and \( 3x + y = 3 \) which is \((-0.5,4.5)\) but since \( x\geq0 \), the relevant vertices are \((0,0)\), \((0,3)\), \((1,0)\) and the part of \( x + y = 4 \) where \( x\geq0,y\geq0 \) but within the \( 3x + y\leq3 \) constraint). The correct graph (among the options) will show this region. If we assume the options are labeled, for example, if one of the options has the shading in the first quadrant, below \( y=-3x + 3 \) and \( y=-x + 4 \), with \( x\geq0 \), that is the correct one. (Since the exact option labels are not fully clear from the image, but the key is the feasible region defined by the two inequalities and non - negativity.)