Graphing Linear Inequalities and Systems - Algebra 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

39. What is the solution set of a system of linear inequalities in two variables?

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

40. What inequality symbol indicates the boundary line is included in the solution set?

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

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

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

42. Which inequality matches shading below a dashed line $y = 3x - 4$?

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

43. What is the first step to graph the inequality $2x + 3y \le 6$?

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

44. Identify the overlap description for the system $x > 1$ and $x < 1$.

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

45. Choose the correct boundary style for $x - 2y \le 0$.

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

46. For $y > mx + b$, which region is shaded relative to the boundary line $y = mx + b$?

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

47. Which side is shaded for $y < -4$?

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

48. Which region is shaded for $y \le \frac{1}{2}x - 3$?

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

49. Identify the solution description for the system $y \ge 1$ and $y \ge 3$.

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

50. Identify the boundary line for the inequality $y \ne 2x + 1$.

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

51. What does it mean if a point is in the solution set of a system of inequalities?

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

52. What does it mean if the shaded regions of a system do not overlap?

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

53. Does the point $(0,0)$ satisfy $y \le -x - 2$?

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

54. Does the point $(0,0)$ satisfy $y > 2x - 1$?

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

55. For the inequality $y \ge 5$, which side of the line $y = 5$ is shaded?

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

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

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

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