This question tests your understanding of graphing linear inequalities and how the solution is represented as a shaded half-plane on the coordinate plane. The solution to a linear inequality is an entire region (a half-plane), not just a single point: every point in the shaded region makes the inequality true! This is different from linear equations, which have just one solution point where the lines cross. To check if point (2, 1) is in the solution region of y < x - 2, we substitute x = 2 and y = 1 into the inequality: 1 < 2 - 2, which simplifies to 1 < 0. Since 1 is NOT less than 0, this statement is false, so the point is not in the solution region. Choice C correctly identifies that the point is not in the solution region because 1 ≮ 0 (1 is not less than 0). Great work! Choices A and B incorrectly claim the point is in the solution (with A making the false claim that 1 < 0), while choice D uses the wrong calculation comparing 1 to -4 instead of 0. For shading direction with y inequalities: y > [line] means 'y is greater than the line' = shade above (higher y-values). y < [line] means 'y is less than the line' = shade below (lower y-values). Or use the test point method: pick (0, 0) if it's not on the line, substitute into the inequality, and if true, shade the side with (0, 0); if false, shade the other side!

This question tests your understanding of graphing linear inequalities and how the solution is represented as a shaded half-plane on the coordinate plane. A system of linear inequalities has a solution region that's the intersection (overlap) of all the individual half-planes: you graph each inequality, and where all the shaded regions overlap is where all the inequalities are satisfied at once. That intersection is your feasible region! For this system, we have three constraints: x ≥ 0 (points on or to the right of the y-axis), y ≥ 0 (points on or above the x-axis), and y ≤ -2x + 6 (points on or below the line y = -2x + 6). The first two constraints together restrict us to the first quadrant (where both x and y are non-negative). The third constraint further restricts us to points below the line y = -2x + 6, with a solid boundary since we have ≤. Choice C correctly identifies this as all points in the first quadrant that are on or below the line y = -2x + 6, including the boundary. Great work! Choice A incorrectly places the region in Quadrant II (where x < 0), Choice B describes the third quadrant (both x ≤ 0 and y ≤ 0), and Choice D has the wrong shading direction (above instead of below). For systems, think of it like finding what's allowed: each inequality restricts the plane, and the solution is where ALL the restrictions are met simultaneously—the overlapping shaded region. If you have y ≥ x and y ≤ -x + 4, the solution is the wedge-shaped region where both shadings overlap!

This question tests your understanding of graphing linear inequalities and how the solution is represented as a shaded half-plane on the coordinate plane. The solution to a linear inequality is an entire region (a half-plane), not just a single point: every point in the shaded region makes the inequality true! This is different from linear equations, which have just one solution point where the lines cross. To check if point (2, 1) is in the solution region of y < x - 2, we substitute x = 2 and y = 1 into the inequality: 1 < 2 - 2, which simplifies to 1 < 0. Since 1 is NOT less than 0, this statement is false, so the point is not in the solution region. Choice C correctly identifies that the point is not in the solution region because 1 ≮ 0 (1 is not less than 0). Great work! Choices A and B incorrectly claim the point is in the solution (with A making the false claim that 1 < 0), while choice D uses the wrong calculation comparing 1 to -4 instead of 0. For shading direction with y inequalities: y > [line] means 'y is greater than the line' = shade above (higher y-values). y < [line] means 'y is less than the line' = shade below (lower y-values). Or use the test point method: pick (0, 0) if it's not on the line, substitute into the inequality, and if true, shade the side with (0, 0); if false, shade the other side!

This question tests your understanding of graphing linear inequalities and how the solution is represented as a shaded half-plane on the coordinate plane. The boundary line for an inequality is the line you'd get if you changed the inequality to equals: for 3x - y ≤ 6, the boundary is 3x - y = 6. The line is dashed for strict inequalities (< or >) because points ON the line don't satisfy the inequality, and solid for ≤ or ≥ because boundary points ARE solutions. To find the boundary line for 3x - y ≤ 6, we simply replace the inequality symbol with an equals sign, giving us 3x - y = 6. Choice A correctly identifies 3x - y = 6 as the boundary line because this is exactly what we get when we change ≤ to =. Great work! Choice B incorrectly keeps the inequality symbol when we need just the equation, choice C has the wrong constant (-6 instead of 6), and choice D changes the minus to plus which alters the equation entirely. For shading direction with y inequalities: y > [line] means 'y is greater than the line' = shade above (higher y-values). y < [line] means 'y is less than the line' = shade below (lower y-values). Or use the test point method: pick (0, 0) if it's not on the line, substitute into the inequality, and if true, shade the side with (0, 0); if false, shade the other side!

This question tests your understanding of graphing linear inequalities and how the solution is represented as a shaded half-plane on the coordinate plane. To graph a linear inequality like y > -3x + 2, we first graph the boundary line y = -3x + 2 (replacing the inequality with equals). Then we decide: is it a solid line (if the inequality includes 'or equal to,' like ≥ or ≤) or a dashed line (if it's strict, like > or <)? Finally, we shade the half-plane that makes the inequality true—above the line for y > or y ≥, below for y < or y ≤. To use the test point method for y > -3x + 2, we substitute (0, 0) into the inequality: 0 > -3(0) + 2, which simplifies to 0 > 2. This statement is FALSE because 0 is not greater than 2. When the test point makes the inequality false, we shade the half-plane that does NOT contain the test point. Choice B correctly states that since 0 > -3(0) + 2 is false, we shade the half-plane that does not contain (0, 0). Great work! Choice A incorrectly evaluates 0 > 2 as true, Choice C has the right evaluation but wrong conclusion about shading, and Choice D has both parts backwards. Or use the test point method: pick (0, 0) if it's not on the line, substitute into the inequality, and if true, shade the side with (0, 0); if false, shade the other side!

This question tests your understanding of graphing linear inequalities and how the solution is represented as a shaded half-plane on the coordinate plane. To graph a linear inequality like y < -3x + 2, we first graph the boundary line y = -3x + 2 (replacing the inequality with equals). Then we decide: is it a solid line (if the inequality includes 'or equal to,' like ≥ or ≤) or a dashed line (if it's strict, like > or <)? Finally, we shade the half-plane that makes the inequality true—above the line for y > or y ≥, below for y < or y ≤. For y < -3x + 2, we need to shade where y-values are less than (smaller than) the expression -3x + 2. Since we want y-values that are smaller, we shade below the boundary line. Choice B correctly identifies shading below the line because y < -3x + 2 means we want all points where the y-coordinate is less than what the line gives us. Great work! Choice A incorrectly shades above (that would be for y > -3x + 2), and choices C and D use left/right language which doesn't apply to non-vertical lines. For shading direction with y inequalities: y > [line] means 'y is greater than the line' = shade above (higher y-values). y < [line] means 'y is less than the line' = shade below (lower y-values). Or use the test point method: pick (0, 0) if it's not on the line, substitute into the inequality, and if true, shade the side with (0, 0); if false, shade the other side!

This question tests your understanding of graphing linear inequalities and how the solution is represented as a shaded half-plane on the coordinate plane. The boundary line for an inequality is the line you'd get if you changed the inequality to equals: for 2x + y ≥ 6, the boundary is 2x + y = 6. The line is dashed for strict inequalities (< or >) because points ON the line don't satisfy the inequality, and solid for ≤ or ≥ because boundary points ARE solutions. To find the boundary line for 2x + y ≥ 6, we simply replace the inequality symbol (≥) with an equals sign (=). This gives us the equation 2x + y = 6, which represents the line that separates the solution region from the non-solution region. Choice B correctly identifies 2x + y = 6 as the boundary line because this is the equation we get when we change the inequality to an equation. Great work! Choices A and C keep the inequality symbols (which don't belong in a boundary line equation), and choice D incorrectly changes the plus to minus. The boundary line for an inequality is always found by replacing the inequality symbol with equals—it's that simple! This line will be drawn as solid (for ≤ or ≥) or dashed (for < or >) depending on whether the boundary points are included in the solution.

This question tests your understanding of graphing linear inequalities and how the solution is represented as a shaded half-plane on the coordinate plane. The boundary line for an inequality is the line you'd get if you changed the inequality to equals: for 2x + y < 6, the boundary is 2x + y = 6. The line is dashed for strict inequalities (< or >) because points ON the line don't satisfy the inequality, and solid for ≤ or ≥ because boundary points ARE solutions. To find the boundary line for 2x + y < 6, we simply replace the inequality symbol with an equals sign, giving us 2x + y = 6. This is the line that separates the plane into two half-planes, one of which will be our solution region. Choice B correctly identifies 2x + y = 6 as the boundary line. Great work! Choice A gives the original inequality, not the boundary line; choice C changes the inequality type but still isn't just the boundary; and choice D incorrectly changes the plus to minus. The boundary line for an inequality is always found by replacing the inequality symbol with equals—it's that simple! Whether the line is drawn solid or dashed depends on the original inequality symbol, but the equation of the boundary line itself is always the equality version.

This question tests your understanding of graphing linear inequalities and how the solution is represented as a shaded half-plane on the coordinate plane. The boundary line for an inequality is the line you'd get if you changed the inequality to equals: for 3x - y ≤ 6, the boundary is 3x - y = 6. The line is dashed for strict inequalities (< or >) because points ON the line don't satisfy the inequality, and solid for ≤ or ≥ because boundary points ARE solutions. To find the boundary line for 3x - y ≤ 6, we simply replace the inequality symbol with an equals sign, giving us 3x - y = 6. Choice A correctly identifies 3x - y = 6 as the boundary line because this is exactly what we get when we change ≤ to =. Great work! Choice B incorrectly keeps the inequality symbol when we need just the equation, choice C has the wrong constant (-6 instead of 6), and choice D changes the minus to plus which alters the equation entirely. For shading direction with y inequalities: y > [line] means 'y is greater than the line' = shade above (higher y-values). y < [line] means 'y is less than the line' = shade below (lower y-values). Or use the test point method: pick (0, 0) if it's not on the line, substitute into the inequality, and if true, shade the side with (0, 0); if false, shade the other side!

This question tests your understanding of graphing linear inequalities and how the solution is represented as a shaded half-plane on the coordinate plane. The boundary line for an inequality is the line you'd get if you changed the inequality to equals: for $2x + y > 6$, the boundary is $2x + y = 6$. The line is dashed for strict inequalities ($<$ or $>$) because points ON the line don't satisfy the inequality, and solid for $≤$ or $≥$ because boundary points ARE solutions. For $2x + y > 5$, the boundary is $2x + y = 5$; to write in slope-intercept form, solve for y: $y = -2x + 5$ (subtract $2x$ from both sides). Choice B correctly identifies $y = -2x + 5$ because solving $2x + y = 5$ for y gives that equation, with slope $-2$ and y-intercept $5$. Great work! Choice A has a positive slope, but it should be negative since it's $+2x$ moving to $-2x$; choice C adds a negative intercept incorrectly; and D keeps the inequality, but the boundary is the equality version—nice try, but remembering to set to equals and solve for y will fix that. The solid-or-dashed rule is simple: if you see $≤$ or $≥$ (the inequality has a line underneath showing 'or equal to'), make the boundary line solid because those points are included. If you see $<$ or $>$ (strict inequality, no line underneath), make it dashed because boundary points don't count. Think: the line under the inequality symbol = solid line on the graph!