Question

a system of inequalities, where x is the rate of a spaceship in a video game and y is where the ship travels on the screen in 3.5 seconds, is represented by the graph. which symbol could be written in both circles in order to represent this system algebraically?

y o 3x
x o –2

options: ≤, ≥, <, >

(graph with x - axis from - 10 to 10, y - axis from - 8 to 10, colored regions, a red line, and a blue vertical line at x = - 2)

Explanation:

Step1: Analyze the line \( y = 3x \)

The line \( y = 3x \) is a solid line (since the shaded region includes the line, so the inequality should be non - strict). The shaded region for \( y\) and \( 3x\) is above the line \( y = 3x\)? Wait, no, let's check the graph. Wait, the blue and red regions: first, for the inequality \( x\circ - 2\), the vertical line \( x=-2\) is a solid line (since the shaded region includes \( x = - 2\)), and the shaded region for \( x\) is to the right of \( x=-2\), so \( x\geq - 2\) (because if it's a solid line, the inequality is non - strict, and the region to the right of \( x=-2\) is \( x\) greater than or equal to \(-2\)). Now for \( y\) and \( 3x\): the line \( y = 3x\) is a solid line (so the inequality is non - strict), and the shaded region for \( y\) and \( 3x\) is above or below? Wait, the slope of \( y = 3x\) is 3. Let's take a test point. If we look at the overlapping region (the purple region), let's see the relationship between \( y\) and \( 3x\). If we consider the line \( y = 3x\), and the shaded region, let's take a point in the purple region, say \( x = 0\), \( y\) values in the purple region at \( x = 0\) are greater than or equal to \( 0\) (since \( y=3x\) at \( x = 0\) is \( 0\)). Wait, no, let's re - examine. The line \( y = 3x\): when \( x = 1\), \( y = 3\). In the blue region (which is part of the solution), at \( x = 1\), \( y\) values are greater than or equal to \( 3\)? Wait, no, the graph: the red line is \( y = 3x\) (since when \( x = 0\), \( y = 0\); when \( x = 1\), \( y = 3\)). The shaded region for \( y\) and \( 3x\) is above the line \( y = 3x\)? Wait, no, the overlapping region (purple) is between \( x\geq - 2\) and the region related to \( y\) and \( 3x\). Wait, the key is the type of line (solid or dashed) and the direction of shading. The line \( x=-2\) is solid, so the inequality for \( x\circ - 2\) is \( x\geq - 2\) (since the shading is to the right of \( x=-2\)). The line \( y = 3x\) is solid, so the inequality for \( y\circ 3x\) should also be a non - strict inequality. Now, let's check the options. The options are \( \leq\), \( \geq\), \( <\), \( >\). Since the lines are solid, we can eliminate \( <\) and \( >\) (because those are for dashed lines). Now, for \( x\circ - 2\): the shading is to the right of \( x=-2\), so \( x\geq - 2\). For \( y\circ 3x\): let's see the slope. The line \( y = 3x\) has a positive slope. The shaded region for \( y\) and \( 3x\): if we take a point in the solution region, say \( (0,0)\) is on the line, and the region above the line? Wait, no, when \( x = 1\), \( y = 3\) is on the line, and the blue region (part of the solution) at \( x = 1\) has \( y\) values greater than or equal to \( 3\)? Wait, no, the graph: the purple region is between \( x\geq - 2\) and the region where \( y\) is greater than or equal to \( 3x\)? Wait, let's think again. The general form of a linear inequality: if the line is solid, the inequality is \( \leq\) or \( \geq\). For the vertical line \( x=-2\), the solution region is \( x\geq - 2\) (shading to the right). For the line \( y = 3x\), the solution region (the part that is in the overall solution) is above the line (since the blue region is above the red line \( y = 3x\) and to the right of \( x=-2\)). So \( y\geq 3x\) (because above the line \( y = 3x\) means \( y\) is greater than or equal to \( 3x\) when the line is solid). And \( x\geq - 2\). So the symbol is \( \geq\).

Step2: Verify the symbol

We check the two inequalities: \( y\geq 3x\) (since the line \( y = 3x\) is solid and the shading is above the line) and \( x\geq - 2\) (since the line \( x=-2\) is solid and the shading is to the right of the line). So the symbol that works for both is \( \geq\).

Answer:

\( \boldsymbol{\geq} \) (corresponding to the option with \( \geq\))