consider the system of inequalities and its graph. \ny ≤ -0.75x \ny ≤ 3x - 2 \nin which section of the graph…

Explanation:

Step1: Analyze the inequalities

We have two inequalities: ( y \leq -0.75x ) (black line) and ( y \leq 3x - 2 ) (red line). The solution to the system is the region that satisfies both inequalities, i.e., the overlap of the regions defined by each inequality.

Step2: Identify the regions

• For ( y \leq -0.75x ), the region is below the black line.
• For ( y \leq 3x - 2 ), the region is below the red line.

Step3: Find the overlap

Looking at the graph, the overlap (the solution region) is the area that is below both lines. From the section numbering (1, 2, 3, 4), we need to check which section is in the overlap. The blue - purple overlapping region (the solution to both inequalities) corresponds to section 4? Wait, no, let's re - check. Wait, the black line is ( y=-0.75x ) (since ( y\leq - 0.75x )) and the red line is ( y = 3x-2 ) (since ( y\leq3x - 2 )). The intersection of the two half - planes (below both lines) is the region that is the solution. Looking at the graph, the section that is in the overlap of the two "below" regions is section 4? Wait, no, maybe I made a mistake. Wait, let's take a test point. Let's take a point in section 4, say (10, - 10). Check ( y\leq - 0.75x ): ( - 10\leq - 0.75\times10=-7.5 ), which is true. Check ( y\leq3x - 2 ): ( - 10\leq3\times10 - 2=28 ), which is true. Now check a point in section 3, say (0, - 5). ( y=-5 ), ( - 0.75x = 0 ), ( - 5\leq0 ) (true for first inequality), ( 3x - 2=-2 ), ( - 5\leq - 2 ) (true). Wait, maybe my initial analysis is wrong. Wait, the red line is ( y = 3x-2 ), when ( x = 0 ), ( y=-2 ). The black line is ( y=-0.75x ), when ( x = 0 ), ( y = 0 ). The solution to the system is the set of points that satisfy both ( y\leq - 0.75x ) and ( y\leq3x - 2 ). So we need to find the region that is below both lines. Let's find the intersection point of the two lines: set ( - 0.75x=3x - 2 ), ( 3x+0.75x = 2 ), ( 3.75x=2 ), ( x=\frac{2}{3.75}=\frac{8}{15}\approx0.533 ), ( y=-0.75\times\frac{8}{15}=-0.4 ). So the two lines intersect at ( (\frac{8}{15},-0.4) ). Now, for ( x <\frac{8}{15} ), which line is lower? Let's take ( x = 0 ), ( y=-0.75x = 0 ), ( y = 3x-2=-2 ). So for ( x <\frac{8}{15} ), ( 3x - 2<-0.75x ), so the lower line is ( y = 3x - 2 ). For ( x>\frac{8}{15} ), ( - 0.75x<3x - 2 ), so the lower line is ( y=-0.75x ). So the solution region is the set of points where ( y\leq\min(-0.75x,3x - 2) ). Looking at the graph, the section that is in the solution (the overlap) is section 4? Wait, maybe the answer is 4. Wait, the options are 1, 2, 3, 4. Let's check the graph again. The blue region is for ( y\leq3x - 2 ), the pink region is for ( y\leq - 0.75x )? No, maybe the blue is ( y\leq3x - 2 ) and the purple is ( y\leq - 0.75x ), and the overlap is the dark blue - purple? Wait, the question is "In which section of the graph does the actual solution to the system lie?". The solution to the system is the intersection of the two solution sets. So we need to find the region that is in both ( y\leq - 0.75x ) and ( y\leq3x - 2 ). So the correct section is 4.

Answer:

4