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Algebra Help: Graphing Linear Inequalities And Systems

Review real example questions for Graphing Linear Inequalities And Systems in Algebra.

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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}

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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)

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 < 3x + 2, graph the boundary y = 3x + 2 (dashed since < is strict), then test (0,0): 0 < 2 is true, so shade the side with (0,0), which is below the line. Choice B correctly identifies shading below the line because y < requires lower y-values than the boundary. If you chose A, that's a common mix-up—gently note that < means below, while > means above; practice with test points to confirm. 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 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!