Algebra Flashcards: Graphing Linear Inequalities And Systems

What this deck covers

This deck focuses on Graphing Linear Inequalities And Systems, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Does the point $(2,1)$ satisfy $y \ge x$?

Answer: No, because $1 \ge 2$ is false. Substitute $(2,1)$: $1 \ge 2$ is false.

Flashcard 2: For the inequality $y < 0$, which side of the line $y = 0$ is shaded?

Answer: Shade below $y = 0$. $y < 0$ means negative $y$-values only.

Flashcard 3: Does the point $(4,2)$ satisfy the system $y > x$ and $y \le 5$?

Answer: No, because $2 > 4$ is false. Check: $2 > 4$ is false, so point fails system.

Flashcard 4: What does it mean if the overlap region of a system is unbounded?

Answer: There are infinitely many solutions extending without end. The solution region extends infinitely in some direction.

Flashcard 5: Which region is shaded for $y > -x + 4$?

Answer: Above the line $y = -x + 4$. $y > -x + 4$ means above the boundary line.

Flashcard 6: Identify the correct boundary style for $y < 2x + 7$.

Answer: Dashed boundary line. Strict inequality ($<$) excludes the boundary.

Flashcard 7: Does the point $(3,-1)$ satisfy $x \le 2$?

Answer: No, because $3 \le 2$ is false. Substitute $(3,-1)$: $3 \le 2$ is false.

Flashcard 8: What is the boundary line for $2x - 5y < 10$?

Answer: The line $2x - 5y = 10$. Set the inequality to equality form.

Flashcard 9: For the inequality $x \le -1$, which side of the line $x = -1$ is shaded?

Answer: Shade to the left of $x = -1$ (including the line). $x \le -1$ includes points with $x$-values $\le -1$.

Flashcard 10: What is the boundary line for the inequality $y > -2$?

Answer: The horizontal line $y = -2$. Horizontal line where all points have $y = -2$.

Flashcard 11: Which side is shaded for $x \ge 2$?

Answer: To the right of $x = 2$ (including the line). $x \ge 2$ includes points with $x$-values $\ge 2$.

Flashcard 12: For $y \le mx + b$, which region is shaded relative to the line $y = mx + b$?

Answer: Shade below the line (including the line). $y \le mx + b$ includes points on and below the line.

Flashcard 13: What boundary is used when graphing the inequality $y \le mx + b$?

Answer: The boundary line $y = mx + b$. The boundary line from the inequality's equation form.

Flashcard 14: What does a dashed boundary line mean when graphing a linear inequality?

Answer: Points on the line are not included in the solution set. The boundary doesn't satisfy the strict inequality.

Flashcard 15: Which boundary line style is used for $y \le mx + b$ or $y \ge mx + b$?

Answer: Solid line (boundary included). Non-strict inequalities include the boundary points.

Flashcard 16: Identify the correct boundary style for $3x - y \ge 1$.

Answer: Solid boundary line. Non-strict inequality ($\ge$) includes the boundary.

Flashcard 17: Which test point should you avoid if the boundary line passes through $(0,0)$?

Answer: Do not use $(0,0)$; choose a different point. $(0,0)$ on the boundary would give $0 = 0$, not helpful.

Flashcard 18: What is a quick method to decide which side of a boundary line to shade?

Answer: Test a point (often $(0,0)$) in the inequality. If the test point satisfies the inequality, shade that side.

Flashcard 19: Find the correct shading for $x + y \le 4$ using test point $(0,0)$.

Answer: Shade the side that contains $(0,0)$. Test $(0,0)$: $0 + 0 \le 4$ is true, so shade that side.

Flashcard 20: Identify the overlap description for the system $y \ge 0$ and $y \le 0$.

Answer: All points on the line $y = 0$. Both inequalities are satisfied only on $y = 0$.

Flashcard 21: What does the solution set of a linear inequality in $x$ and $y$ represent on a coordinate plane?

Answer: A half-plane (all points that satisfy the inequality). It includes all points making the inequality true.

Flashcard 22: Identify the solution description for the system $x \le 2$ and $x \le -1$.

Answer: All points with $x \le -1$. The more restrictive condition $x \le -1$ determines the solution.

Flashcard 23: For $y \ge mx + b$, which region is shaded relative to the line $y = mx + b$?

Answer: Shade above the line (including the line). $y \ge mx + b$ includes points on and above the line.

Flashcard 24: For $y < mx + b$, which region is shaded relative to the boundary line $y = mx + b$?

Answer: Shade below the line. $y < mx + b$ means $y$-values less than the line.

Flashcard 25: What does a solid boundary line mean when graphing a linear inequality?

Answer: Points on the line are included in the solution set. The boundary satisfies the inequality condition.

Flashcard 26: Does the point $(-2,5)$ satisfy $x > -3$?

Answer: Yes, because $-2 > -3$ is true. Substitute $(-2,5)$: $-2 > -3$ is true.

Flashcard 27: Which boundary line style is used for $y < mx + b$ or $y > mx + b$?

Answer: Dashed line (boundary not included). Strict inequalities exclude the boundary points.

Flashcard 28: Does the point $(0,5)$ satisfy the system $y \ge 2x + 1$ and $y < 6$?

Answer: Yes, because $5 \ge 1$ and $5 < 6$. Check: $5 \ge 1$ and $5 < 6$ are both true.

Flashcard 29: Choose the correct boundary style for $x + y > 8$.

Answer: Dashed line. Strict inequality ($>$) excludes the boundary.

Flashcard 30: Does the point $(-1,3)$ satisfy $y < 2x + 6$?

Answer: Yes, because $3 < 4$ is true. Substitute $(-1,3)$: $3 < 2(-1) + 6 = 4$ is true.

Flashcard 31: Does the point $(1,2)$ satisfy the system $y \ge x$ and $y \le 3$?

Answer: Yes, because $2 \ge 1$ and $2 \le 3$. Check: $2 \ge 1$ and $2 \le 3$ are both true.

Flashcard 32: What is the boundary line for the inequality $x \le 3$?

Answer: The vertical line $x = 3$. Vertical line where all points have $x = 3$.

Flashcard 33: Which inequality matches shading above a solid line $y = -2x + 1$?

Answer: $y \ge -2x + 1$. Solid line with above shading indicates $\ge$.

Flashcard 34: What is the boundary line for $y \ge -3x + 2$?

Answer: The line $y = -3x + 2$. Set the inequality to equality form.

Flashcard 35: For the inequality $x > 4$, which side of the line $x = 4$ is shaded?

Answer: Shade to the right of $x = 4$. $x > 4$ means $x$-values greater than 4.

Flashcard 36: What is a standard test point for shading when the boundary line does not pass through $(0,0)$?

Answer: Use $(0,0)$ as the test point. Origin is convenient when not on the boundary line.

Flashcard 37: What inequality symbol indicates the boundary line is excluded from the solution set?

Answer: Strict symbols: $<$ or $>$. Strict symbols create open boundaries (dashed lines).

Flashcard 38: Find the correct shading for $2x + 3y \ge 6$ using test point $(0,0)$.

Answer: Shade the side that does not contain $(0,0)$. Test $(0,0)$: $0 + 0 \ge 6$ is false, so shade opposite.

Flashcard 39: Which side is shaded for $y < -4$?

Answer: Below $y = -4$. $y < -4$ means $y$-values less than $-4$.

Flashcard 40: What is the solution set of a system of linear inequalities in two variables?

Answer: The intersection of the half-planes for all inequalities. Only points satisfying all inequalities simultaneously.

Flashcard 41: What inequality symbol indicates the boundary line is included in the solution set?

Answer: Inclusive symbols: $\le$ or $\ge$. Inclusive symbols create closed boundaries (solid lines).

Flashcard 42: What is the intersection meaning in a system: points that satisfy inequality $1$ and inequality $2$?

Answer: Points that satisfy both inequalities at the same time. Intersection means 'and' - both conditions must hold.

Flashcard 43: Which inequality matches shading below a dashed line $y = 3x - 4$?

Answer: $y < 3x - 4$. Dashed line with below shading indicates $<$.

Flashcard 44: What is the first step to graph the inequality $2x + 3y \le 6$?

Answer: Graph the boundary line $2x + 3y = 6$. Draw the boundary by setting the inequality to equality.

Flashcard 45: Identify the overlap description for the system $x > 1$ and $x < 1$.

Answer: No solution (empty intersection). No $x$-value can be both $> 1$ and $< 1$.

Flashcard 46: Choose the correct boundary style for $x - 2y \le 0$.

Answer: Solid line. Non-strict inequality ($\le$) includes the boundary.

Flashcard 47: For $y > mx + b$, which region is shaded relative to the boundary line $y = mx + b$?

Answer: Shade above the line. $y > mx + b$ means $y$-values greater than the line.

Flashcard 48: Which side is shaded for $y < -4$?

Answer: Below $y = -4$. $y < -4$ means $y$-values less than $-4$.

Flashcard 49: Which region is shaded for $y \le \frac{1}{2}x - 3$?

Answer: Below the line $y = \frac{1}{2}x - 3$ (including it). $y \le \frac{1}{2}x - 3$ includes on and below the line.

Flashcard 50: Identify the solution description for the system $y \ge 1$ and $y \ge 3$.

Answer: All points with $y \ge 3$. The more restrictive condition $y \ge 3$ determines the solution.

Flashcard 51: Identify the boundary line for the inequality $y \ne 2x + 1$.

Answer: Not a linear inequality; $y \ne 2x + 1$ is not half-plane form. $\ne$ creates two regions, not a single half-plane.

Flashcard 52: What does it mean if a point is in the solution set of a system of inequalities?

Answer: It satisfies every inequality in the system. The point makes all system inequalities true.

Flashcard 53: What does it mean if the shaded regions of a system do not overlap?

Answer: The system has no solution (empty intersection). No points satisfy all inequalities simultaneously.

Flashcard 54: Does the point $(0,0)$ satisfy $y \le -x - 2$?

Answer: No, because $0 \le -2$ is false. Substitute $(0,0)$: $0 \le 0 - 2 = -2$ is false.

Flashcard 55: Does the point $(0,0)$ satisfy $y > 2x - 1$?

Answer: Yes, because $0 > -1$ is true. Substitute $(0,0)$: $0 > 2(0) - 1 = -1$ is true.

Flashcard 56: For the inequality $y \ge 5$, which side of the line $y = 5$ is shaded?

Answer: Shade above $y = 5$ (including the line). $y \ge 5$ includes points with $y$-values $\ge 5$.

Flashcard 57: Does the point $(2,6)$ satisfy the system $y \ge 2x + 1$ and $y < 6$?

Answer: No, because $6 < 6$ is false. Check: $6 < 6$ is false, so point fails system.