Graphing Linear Inequalities and Systems - Algebra
Card 1 of 30
Does the point $(1,2)$ satisfy the system $y \ge x$ and $y \le 3$?
Does the point $(1,2)$ satisfy the system $y \ge x$ and $y \le 3$?
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Yes, because $2 \ge 1$ and $2 \le 3$. Check: $2 \ge 1$ and $2 \le 3$ are both true.
Yes, because $2 \ge 1$ and $2 \le 3$. Check: $2 \ge 1$ and $2 \le 3$ are both true.
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Identify the overlap description for the system $x > 1$ and $x < 1$.
Identify the overlap description for the system $x > 1$ and $x < 1$.
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No solution (empty intersection). No $x$-value can be both $> 1$ and $< 1$.
No solution (empty intersection). No $x$-value can be both $> 1$ and $< 1$.
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Identify the overlap description for the system $y \ge 0$ and $y \le 0$.
Identify the overlap description for the system $y \ge 0$ and $y \le 0$.
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All points on the line $y = 0$. Both inequalities are satisfied only on $y = 0$.
All points on the line $y = 0$. Both inequalities are satisfied only on $y = 0$.
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Which side is shaded for $y < -4$?
Which side is shaded for $y < -4$?
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Below $y = -4$. $y < -4$ means $y$-values less than $-4$.
Below $y = -4$. $y < -4$ means $y$-values less than $-4$.
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Choose the correct boundary style for $x + y > 8$.
Choose the correct boundary style for $x + y > 8$.
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Dashed line. Strict inequality ($>$) excludes the boundary.
Dashed line. Strict inequality ($>$) excludes the boundary.
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What does it mean if the shaded regions of a system do not overlap?
What does it mean if the shaded regions of a system do not overlap?
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The system has no solution (empty intersection). No points satisfy all inequalities simultaneously.
The system has no solution (empty intersection). No points satisfy all inequalities simultaneously.
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What does it mean if the overlap region of a system is unbounded?
What does it mean if the overlap region of a system is unbounded?
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There are infinitely many solutions extending without end. The solution region extends infinitely in some direction.
There are infinitely many solutions extending without end. The solution region extends infinitely in some direction.
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For $y > mx + b$, which region is shaded relative to the boundary line $y = mx + b$?
For $y > mx + b$, which region is shaded relative to the boundary line $y = mx + b$?
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Shade above the line. $y > mx + b$ means $y$-values greater than the line.
Shade above the line. $y > mx + b$ means $y$-values greater than the line.
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Does the point $(-2,5)$ satisfy $x > -3$?
Does the point $(-2,5)$ satisfy $x > -3$?
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Yes, because $-2 > -3$ is true. Substitute $(-2,5)$: $-2 > -3$ is true.
Yes, because $-2 > -3$ is true. Substitute $(-2,5)$: $-2 > -3$ is true.
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Does the point $(3,-1)$ satisfy $x \le 2$?
Does the point $(3,-1)$ satisfy $x \le 2$?
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No, because $3 \le 2$ is false. Substitute $(3,-1)$: $3 \le 2$ is false.
No, because $3 \le 2$ is false. Substitute $(3,-1)$: $3 \le 2$ is false.
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Which side is shaded for $x \ge 2$?
Which side is shaded for $x \ge 2$?
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To the right of $x = 2$ (including the line). $x \ge 2$ includes points with $x$-values $\ge 2$.
To the right of $x = 2$ (including the line). $x \ge 2$ includes points with $x$-values $\ge 2$.
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Choose the correct boundary style for $x - 2y \le 0$.
Choose the correct boundary style for $x - 2y \le 0$.
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Solid line. Non-strict inequality ($\le$) includes the boundary.
Solid line. Non-strict inequality ($\le$) includes the boundary.
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What is the boundary line for $2x - 5y < 10$?
What is the boundary line for $2x - 5y < 10$?
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The line $2x - 5y = 10$. Set the inequality to equality form.
The line $2x - 5y = 10$. Set the inequality to equality form.
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Does the point $(2,6)$ satisfy the system $y \ge 2x + 1$ and $y < 6$?
Does the point $(2,6)$ satisfy the system $y \ge 2x + 1$ and $y < 6$?
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No, because $6 < 6$ is false. Check: $6 < 6$ is false, so point fails system.
No, because $6 < 6$ is false. Check: $6 < 6$ is false, so point fails system.
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Identify the solution description for the system $x \le 2$ and $x \le -1$.
Identify the solution description for the system $x \le 2$ and $x \le -1$.
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All points with $x \le -1$. The more restrictive condition $x \le -1$ determines the solution.
All points with $x \le -1$. The more restrictive condition $x \le -1$ determines the solution.
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Which boundary line style is used for $y \le mx + b$ or $y \ge mx + b$?
Which boundary line style is used for $y \le mx + b$ or $y \ge mx + b$?
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Solid line (boundary included). Non-strict inequalities include the boundary points.
Solid line (boundary included). Non-strict inequalities include the boundary points.
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What is the boundary line for the inequality $x \le 3$?
What is the boundary line for the inequality $x \le 3$?
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The vertical line $x = 3$. Vertical line where all points have $x = 3$.
The vertical line $x = 3$. Vertical line where all points have $x = 3$.
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What does it mean if a point is in the solution set of a system of inequalities?
What does it mean if a point is in the solution set of a system of inequalities?
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It satisfies every inequality in the system. The point makes all system inequalities true.
It satisfies every inequality in the system. The point makes all system inequalities true.
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What is the solution set of a system of linear inequalities in two variables?
What is the solution set of a system of linear inequalities in two variables?
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The intersection of the half-planes for all inequalities. Only points satisfying all inequalities simultaneously.
The intersection of the half-planes for all inequalities. Only points satisfying all inequalities simultaneously.
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What does the solution set of a linear inequality in $x$ and $y$ represent on a coordinate plane?
What does the solution set of a linear inequality in $x$ and $y$ represent on a coordinate plane?
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A half-plane (all points that satisfy the inequality). It includes all points making the inequality true.
A half-plane (all points that satisfy the inequality). It includes all points making the inequality true.
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What boundary is used when graphing the inequality $y \le mx + b$?
What boundary is used when graphing the inequality $y \le mx + b$?
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The boundary line $y = mx + b$. The boundary line from the inequality's equation form.
The boundary line $y = mx + b$. The boundary line from the inequality's equation form.
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Which boundary line style is used for $y < mx + b$ or $y > mx + b$?
Which boundary line style is used for $y < mx + b$ or $y > mx + b$?
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Dashed line (boundary not included). Strict inequalities exclude the boundary points.
Dashed line (boundary not included). Strict inequalities exclude the boundary points.
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For $y \le mx + b$, which region is shaded relative to the line $y = mx + b$?
For $y \le mx + b$, which region is shaded relative to the line $y = mx + b$?
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Shade below the line (including the line). $y \le mx + b$ includes points on and below the line.
Shade below the line (including the line). $y \le mx + b$ includes points on and below the line.
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For $y < mx + b$, which region is shaded relative to the boundary line $y = mx + b$?
For $y < mx + b$, which region is shaded relative to the boundary line $y = mx + b$?
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Shade below the line. $y < mx + b$ means $y$-values less than the line.
Shade below the line. $y < mx + b$ means $y$-values less than the line.
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For $y \ge mx + b$, which region is shaded relative to the line $y = mx + b$?
For $y \ge mx + b$, which region is shaded relative to the line $y = mx + b$?
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Shade above the line (including the line). $y \ge mx + b$ includes points on and above the line.
Shade above the line (including the line). $y \ge mx + b$ includes points on and above the line.
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Which test point should you avoid if the boundary line passes through $(0,0)$?
Which test point should you avoid if the boundary line passes through $(0,0)$?
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Do not use $(0,0)$; choose a different point. $(0,0)$ on the boundary would give $0 = 0$, not helpful.
Do not use $(0,0)$; choose a different point. $(0,0)$ on the boundary would give $0 = 0$, not helpful.
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What is a quick method to decide which side of a boundary line to shade?
What is a quick method to decide which side of a boundary line to shade?
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Test a point (often $(0,0)$) in the inequality. If the test point satisfies the inequality, shade that side.
Test a point (often $(0,0)$) in the inequality. If the test point satisfies the inequality, shade that side.
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What does a dashed boundary line mean when graphing a linear inequality?
What does a dashed boundary line mean when graphing a linear inequality?
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Points on the line are not included in the solution set. The boundary doesn't satisfy the strict inequality.
Points on the line are not included in the solution set. The boundary doesn't satisfy the strict inequality.
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What does a solid boundary line mean when graphing a linear inequality?
What does a solid boundary line mean when graphing a linear inequality?
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Points on the line are included in the solution set. The boundary satisfies the inequality condition.
Points on the line are included in the solution set. The boundary satisfies the inequality condition.
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Does the point $(-1,3)$ satisfy $y < 2x + 6$?
Does the point $(-1,3)$ satisfy $y < 2x + 6$?
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Yes, because $3 < 4$ is true. Substitute $(-1,3)$: $3 < 2(-1) + 6 = 4$ is true.
Yes, because $3 < 4$ is true. Substitute $(-1,3)$: $3 < 2(-1) + 6 = 4$ is true.
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