Question 1: Number lines are graphical tools used to represent inequalities by marking the range of possible solutions. Correct plotting involves accurately placing boundary points and using appropriate shading or symbols to indicate the solution set.

Answer: Number lines visually show solutions to inequalities, with correct placement of boundaries and shading reflecting the inequality sign.

Question 1: Plot the inequality y > 3 on the number line. Use an open circle at 3 and shade the region greater than 3.

Answer: Correctly plotted open circle at 3, shading to the right.

Question 2: Place a solid circle at -2 and shade the region less than or equal to -2 for the inequality x ≤ -2.

Answer: Solid circle at -2, shading to the left.

Question 3: Identify the mistake in plotting the inequality y < 1 if the circle is filled and shading is to the right.

Answer: Incorrect: For y < 1, the circle should be open, and shading should be to the left.

Question 4: A student plots x ≥ 4 but uses a dashed circle at 4 and shades the region to the left. Explain the errors and correct the plot.

Answer: Error: For x ≥ 4, the circle should be solid, and shading should be to the right. Correction: solid circle at 4, shading to the right.

Question 5: On a number line, how would you represent the solution set of the inequality -1 < x < 3?

Answer: Open circles at -1 and 3, shading between them.

Question 6: A graph shows a solid circle at 0 with shading to the right. Which inequality does this correctly represent?

Answer: x ≥ 0

Question 7: Construct a number line to represent the inequality x < -2 or x > 2. Use appropriate symbols and shading.

Answer: Open circles at -2 and 2, shading to the left of -2 and to the right of 2, representing the union.

Question 8: Identify the misconception in the following plot: a solid circle at 5 with shading to the left, labeled as x ≥ 5.

Answer: Misconception: For x ≥ 5, the circle should be solid, and shading should be to the right, not left.

Question 9: Evaluate the correctness of plotting y ≤ 2 with an open circle at 2 and shading above 2.

Answer: Incorrect: For y ≤ 2, the circle should be solid, and shading below 2.

Question 10: Given the inequality x > -1, sketch the number line and describe the key features to indicate the solution set.

Answer: Open circle at -1, shading to the right, indicating all x greater than -1.

Question 11: A student incorrectly shades the region between -3 and 2 with both endpoints open and labels it as x ≠ -3 and x ≠ 2. What is the mistake and how should it be corrected?

Answer: Mistake: The shading suggests solutions between -3 and 2, but the inequality x ≠ -3 and x ≠ 2 does not specify an interval. Correct: Plot the inequality as x > -3 and x < 2 with open circles at -3 and 2.