Algebra Quiz: Graphing Linear Inequalities And Systems

20 questions · exam conditions

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Graphing Linear Inequalities And SystemsQuestion 1 of 20

A small theater must follow these constraints for ticket sales: $x$ = adult tickets, $y$ = student tickets. $$ \begin{cases} x + y \le 80 \ x \ge 10 \ y \ge 15 \end{cases}

Points with

x\le 10

y\le 15

, and

x+y\ge 80

Points with

x\ge 10

y\ge 15

, and

x+y\le 80

(intersection of all three half-planes)

Points with

x\ge 10

y\ge 15

x+y\le 80

(union of the half-planes)

Points with

x\le 10

y\ge 15

, and

x+y\le 80

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Algebra Quiz

Algebra Quiz: Graphing Linear Inequalities And Systems

Practice Graphing Linear Inequalities And Systems in Algebra with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Graphing Linear Inequalities And Systems, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A small theater must follow these constraints for ticket sales: $x$ = adult tickets, $y$ = student tickets. $$ \begin{cases} x + y \le 80 \ x \ge 10 \ y \ge 15 \end{cases}

Points with $x\le 10$ , $y\le 15$ , and $x+y\ge 80$
Points with $x\ge 10$ , $y\ge 15$ , and $x+y\le 80$ (intersection of all three half-planes) (correct answer)
Points with $x\ge 10$ or $y\ge 15$ or $x+y\le 80$ (union of the half-planes)
Points with $x\le 10$ , $y\ge 15$ , and $x+y\le 80$

Explanation: 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 x + y ≤ 80 (solid line x+y=80, shade below), x ≥ 10 (solid vertical, shade right), y ≥ 15 (solid horizontal, shade above), the feasible region is the polygon where all overlap: x≥10, y≥15, x+y≤80. Choice B correctly identifies points with x≥10, y≥15, and x+y≤80 as the intersection of all three half-planes. Choice C says 'or' which is union, but systems require all conditions (intersection)—great effort, but remember 'and' for systems. 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!

Question 2

Is point $(2,1)$ in the solution region of the inequality $y> x-1$ ?

Yes, because $1>2-1$ is true.
Yes, because $1\ge 2-1$ is true.
No, because $1>2-1$ is false. (correct answer)
No, because $1\le 2-1$ is true.

Explanation: 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 - 1, we substitute x = 2 and y = 1 into the inequality. This gives us: 1 > 2 - 1, which simplifies to 1 > 1. Since 1 is not greater than 1 (they're equal), this statement is false. Choice C correctly states that (2, 1) is not in the solution region because 1 > 2 - 1 is false. Great work! Choice A incorrectly evaluates the arithmetic (2 - 1 = 1, not something less than 1), Choice B uses ≥ instead of the given >, and Choice D introduces ≤ which isn't part of the original inequality. 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!

Question 3

Should the boundary line for the inequality $y \le -x + 4$ be solid or dashed?

Dashed, because the inequality is strict
Solid, because the inequality includes equality (correct answer)
Dashed, because the slope is negative
Solid, because $y$ is isolated on the left side

Explanation: 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 y ≤ -x + 4, the boundary is y = -x + 4, and since it includes 'or equal to' (≤), we draw a solid line starting from points like (0,4) and (4,0); the shading would be below the line because y is less than or equal to the line. Choice B correctly identifies the line as solid because the inequality includes equality with the ≤ symbol. If you picked A or C, you might have confused strict inequalities with this one—gently remember that the presence of 'or equal to' always means solid, regardless of slope or form. 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!

Question 4

What is the boundary line for the linear inequality $3x - y \le 6$ ?

$y = 3x + 6$
$3x - y = 6$ (correct answer)
$3x + y = 6$
$3x - y < 6$

Explanation: 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 3x - y ≤ 6, replace the inequality with equals to get 3x - y = 6; this is the boundary line, and since it includes equality (≤), it would be solid when graphed, but the question asks only for the boundary equation. Choice B correctly identifies 3x - y = 6 because it directly replaces the inequality symbol with equals while keeping the terms the same. Great work! Some might rewrite it in slope-intercept form like choice D, but that's not necessary—the boundary is the equality version of the given inequality, so choices A, C, and D alter it incorrectly; don't worry, reviewing the definition will help clarify 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!

Question 5

Which direction should be shaded for the inequality $y < 3x + 2$ (using the boundary line $y=3x+2$ )?

Shade to the right of the line
Shade to the left of the line
Shade above the line
Shade below the line (correct answer)

Question 6

Which direction should be shaded for the inequality $y<-3x+2$ ?

Shade above the line $y=-3x+2$ .
Shade below the line $y=-3x+2$ . (correct answer)
Shade to the right of the line $y=-3x+2$ .
Shade to the left of the line $y=-3x+2$ .

Explanation: 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!

Question 7

What is the boundary line for the inequality $2x + y \ge 6$ ?

$y = 2x + 6$
$2x + y = 6$ (correct answer)
$2x - y = 6$
$2x + y \ge 6$

Explanation: 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 ≥ 6, the boundary is simply 2x + y = 6, which you can graph by finding intercepts (x=3 when y=0, y=6 when x=0) and drawing a solid line since ≥ includes equality. Choice B correctly identifies the boundary as 2x + y = 6 because it's the equality version of the inequality. Choice A includes the inequality symbol, but remember, the boundary is just the line itself—nice try, but focus on replacing the inequality with equals. 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!

Question 8

What is the boundary line for the inequality $2x+y\ge 6$ ?

$2x+y>6$
$2x+y=6$ (correct answer)
$2x+y\le 6$
$2x-y=6$

Explanation: 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.

Question 9

What is the boundary line for the inequality $x-2y<8$ ?

$x-2y=8$ (correct answer)
$x-2y\le 8$
$x+2y=8$
$x-2y<8$

Explanation: 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 x - 2y < 8, the boundary is x - 2y = 8. 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 x - 2y < 8, we replace the inequality symbol (<) with an equals sign (=). This gives us x - 2y = 8, which is the line that separates the solution region from the non-solution region. Choice A correctly identifies x - 2y = 8 as the boundary line because this is what we get when we change the inequality to an equation. Great work! Choices B and D keep inequality symbols (which don't belong in a boundary line equation), and choice C incorrectly changes the minus to plus. The boundary line for an inequality is always found by replacing the inequality symbol with equals—it's that simple! Remember, the boundary line is an equation (with =), not an inequality, regardless of whether it will be drawn solid or dashed.

Question 10

Which direction should be shaded for the inequality $y\le -x+4$ ?

Shade below the line $y=-x+4$ . (correct answer)
Shade to the right of the line $y=-x+4$ .
Shade above the line $y=-x+4$ .
Shade to the left of the line $y=-x+4$ .

Explanation: 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 ≤ -x + 4, we first graph the boundary line y = -x + 4 (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 ≤ -x + 4, the boundary line is y = -x + 4 (a line with slope -1 and y-intercept 4). Since we have ≤, the line is solid. The key is determining shading: y ≤ -x + 4 means y is less than or equal to the expression -x + 4, so we shade below the line where y-values are smaller. Choice B correctly identifies that we should shade below the line y = -x + 4. Great work! Choice A incorrectly shades above (that would be for y ≥ -x + 4), while choices C and D use 'right' and 'left' which don't apply well to non-vertical lines—we use 'above' and 'below' for most linear inequalities. 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!

Question 11

What is the boundary line for the inequality $2x-y>3$ ?

$2x-y\ge 3$
$2x-y=3$ (correct answer)
$2x-y<3$
$2x-y\le 3$

Explanation: 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 > 3, the boundary is 2x - y = 3. 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 > 3, we simply replace the inequality symbol (>) with an equals sign (=). This gives us the equation 2x - y = 3, which represents the line that separates the plane into two half-planes. Choice B correctly identifies 2x - y = 3 as the boundary line. Great work! Choices A, C, and D incorrectly keep inequality symbols—remember, a boundary line is always an equation (with =), not an inequality. The boundary line itself will be dashed when graphed because we have a strict inequality (>). 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!

Question 12

A system of three linear inequalities creates a triangular solution region. If one vertex of this triangle is at $(2, 5)$ and the three boundary lines are $y = x + 3$ , $y = -2x + 9$ , and $x = 0$ , what type of boundary does the line $x = 0$ have in this system?

Solid boundary with inequality $x \geq 0$ to create the triangular region (correct answer)
Dashed boundary with inequality $x > 0$ to create the triangular region
Solid boundary with inequality $x \leq 0$ to create the triangular region
The boundary type cannot be determined without additional vertex information

Explanation: First, verify (2,5) is at intersection of two lines: y = x + 3 gives 5 = 2 + 3 ✓. y = -2x + 9 gives 5 = -4 + 9 ✓. So (2,5) is where these lines intersect. For a triangular region in the first quadrant with vertex at (2,5), the third boundary x = 0 must use x ≥ 0 to include the region to the right of the y-axis. The boundary should be solid since we need the triangle to have definite vertices. Choice B: Dashed boundary would exclude the y-axis. Choice C: x ≤ 0 would put triangle in second quadrant. Choice D: We have sufficient information to determine this.

Question 13

Is point $(0,0)$ in the solution region of the system $$ \begin{cases} y>2x-3 \ y\le x+1 \end{cases}

Yes, because it satisfies both inequalities. (correct answer)
No, because it fails $y>2x-3$ only.
No, because it fails $y\le x+1$ only.
No, because it fails both inequalities.

Explanation: 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! For a system, a point must satisfy ALL inequalities to be in the solution region. To check if (0, 0) is in the solution region, we test both inequalities. For y > 2x - 3: substituting gives 0 > 2(0) - 3, which is 0 > -3, TRUE. For y ≤ x + 1: substituting gives 0 ≤ 0 + 1, which is 0 ≤ 1, TRUE. Since both inequalities are satisfied, the point is in the solution region. Choice A correctly identifies that (0, 0) is in the solution region because it satisfies both inequalities: 0 > -3 and 0 ≤ 1 are both true statements. Great work! Choices B, C, and D incorrectly claim the point fails one or both inequalities when our calculations show both are satisfied. 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! For systems, the point must satisfy ALL inequalities—if even one fails, the point is outside the solution region.

Question 14

Is point $(2,1)$ in the solution region of the inequality $y < x - 2$ ?

Yes, because $1 < 0$
Yes, because $1 < 4$
No, because $1 \not< 0$ (correct answer)
No, because $1 \not< -4$

Explanation: 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!

Question 15

Graph the solution set to the system $$ \begin{cases} x \ge 0 \ y \ge 0 \ y \le -x + 4 \end{cases}

All points above the line $y = -x + 4$ in any quadrant
All points in Quadrant II that are above the line $y = -x + 4$
All points in the first quadrant that are on or below the line $y = -x + 4$ (correct answer)
All points in the first quadrant that are on or above the line $y = -x + 4$

x \ge 0

(solid vertical at x=0, shade right),

y \ge 0

(solid horizontal at y=0, shade above),

y \le -x + 4

(solid line, shade below or on), the overlap is the first quadrant triangle on or below

y = -x + 4

. Choice B correctly identifies all points in the first quadrant that are on or below the line

y = -x + 4

because it captures the intersection in the non-negative region. Choice A mentions Quadrant II, but that's x<0—gently confirm quadrants: first is

x \ge 0

y \ge 0

. 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 \ge x

and

y \le -x + 4

, the solution is the wedge-shaped region where both shadings overlap!

Question 16

Is point $(2,1)$ in the solution region of the inequality $y\ge x-1$ ?

Yes, because $1\ge 2-1$ is true. (correct answer)
No, because $1\ge 2-1$ is false.
Yes, because $(2,1)$ is below the line $y=x-1$ .
No, because the boundary line is dashed.

Explanation: 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 - 1, we substitute x = 2 and y = 1 into the inequality: 1 ≥ 2 - 1, which simplifies to 1 ≥ 1. Since this is true (1 equals 1, and ≥ includes equality), the point is in the solution region. Choice A correctly identifies that (2, 1) is in the solution region because 1 ≥ 2 - 1 evaluates to 1 ≥ 1, which is true. Great work! Choice B incorrectly claims this is false, choice C gives a wrong geometric reason (the point is actually on the line, not below it), and choice D is irrelevant since the boundary would be solid for ≥. To test if a point is in the solution region, simply substitute its coordinates into the inequality. If the resulting statement is true, the point is in the solution region. If false, it's not. This works for any point and any linear inequality!

Question 17

Graph the inequality $y > 2x - 1$ . Which description is correct for the boundary line and the shaded half-plane (solution region)?

Solid boundary line $y=2x-1$ ; shade above the line
Dashed boundary line $y=-2x-1$ ; shade above the line
Dashed boundary line $y=2x-1$ ; shade above the line (correct answer)
Solid boundary line $y=2x-1$ ; shade below the line

Explanation: 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 > 2x + 1, we first graph the boundary line y = 2x + 1 (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 > 2x - 1, start by graphing the boundary line y = 2x - 1 using points like (0, -1) and (1, 1); since it's a strict inequality (>), make the line dashed, then test a point like (0,0) in y > 2x - 1: 0 > -1 is true, so shade the side containing (0,0), which is above the line. Choice B correctly identifies the dashed boundary line y=2x-1 with shading above the line because the inequality is strict and requires y-values greater than the line. A common mistake is choosing a solid line like in choice A or D, but remember, strict inequalities use dashed lines to show boundary points aren't included—keep practicing to spot that difference! 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! 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!

Question 18

Which inequality represents the shaded region shown in the coordinate plane below?

$2x - 3y > 6$ and includes the boundary line
$2x - 3y < 6$ and excludes the boundary line (correct answer)
$2x - 3y \geq 6$ and includes the boundary line
$2x - 3y \leq 6$ and excludes the boundary line

Explanation: The boundary line passes through (3, 0) and (0, -2), giving slope 2/3 and equation 2x - 3y = 6. The shaded region is below and to the left of this line. Testing the origin (0, 0): 2(0) - 3(0) = 0 < 6, so the shaded region satisfies 2x - 3y < 6. The dashed line indicates the boundary is not included. Choice A has wrong inequality direction and wrong boundary. Choice C has wrong inequality direction. Choice D has correct inequality but wrong boundary inclusion.

Question 19

What is the boundary line for the inequality $2x + y > 5$ (written in slope-intercept form)?

$y = -2x - 5$
$y = 2x + 5$
$y > -2x + 5$
$y = -2x + 5$ (correct answer)

2x + y > 6

, the boundary is

2x + y = 6

. The line is dashed for strict inequalities (

<

>

) because points ON the line don't satisfy the inequality, and solid for

≤

≥

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

≤

≥

(the inequality has a line underneath showing 'or equal to'), make the boundary line solid because those points are included. If you see

<

>

(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!

Question 20

Refer to the system of inequalities graphed below. A new constraint $y \geq x - 2$ is added to this existing system. How does this change the solution region?

The solution region expands to include additional area above the new line
The solution region contracts as the new constraint eliminates part of the original region (correct answer)
The solution region remains unchanged because the new constraint is redundant
The solution region becomes empty because the constraints are contradictory

Explanation: Adding a new constraint to a system of inequalities can only maintain or reduce the solution region - it cannot expand it. The new constraint y ‚â• x - 2 creates an additional requirement that must be satisfied. Any points in the original solution region that don't satisfy y ‚â• x - 2 will be eliminated from the new solution region. Choice A: Adding constraints cannot expand the solution region. Choice C: The constraint would only be redundant if all points in the original region already satisfied y ‚â• x - 2. Choice D: While possible, this would require checking if the new constraint is compatible with the existing ones.