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Write down the inequalities that are represented by the following diagrams.

Mathematics

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Here are the inequalities for Question 3 and the solution for Question 4:

Write down the inequalities that are represented by the following diagrams.

A)

  1. Identify the first part of the inequality.

    There is an open circle at 11 with shading to the left, which means x<11x < 11.

  2. Identify the second part of the inequality.

    There is a solid circle at 18 with shading to the right, which means x≥18x \ge 18.

  3. Combine the inequalities with "OR" since they are separate regions.

    x<11 OR x≥18\boxed{x < 11 \text{ OR } x \ge 18}

    B)

  4. Identify the left boundary.

    There is a solid circle at -2 with shading to the right, which means x≥−2x \ge -2.

  5. Identify the right boundary.

    There is an open circle at 6 with shading to the left, which means x<6x < 6.

  6. Combine the inequalities with "AND" since the region is between the two points.

    −2≤x<6\boxed{-2 \le x < 6}

    C)

  7. Identify the left boundary.

    There is a solid circle at 0 with shading to the right, which means x≥0x \ge 0.

  8. Identify the right boundary.

    There is an open circle at 16 with shading to the left, which means x<16x < 16.

  9. Combine the inequalities with "AND" since the region is between the two points.

    0≤x<16\boxed{0 \le x < 16}

    D)

  10. Identify the first part of the inequality.

    There is an open circle at 5 with shading to the left, which means x<5x < 5.

  11. Identify the second part of the inequality.

    There is an open circle at 15 with shading to the right, which means x>15x > 15.

  12. Combine the inequalities with "OR" since they are separate regions.

    x<5 OR x>15\boxed{x < 5 \text{ OR } x > 15}

    E)

  13. Identify the left boundary.

    There is a solid circle at -2 with shading to the right, which means x≥−2x \ge -2.

  14. Identify the right boundary.

    There is a solid circle at 4 with shading to the left, which means x≤4x \le 4.

  15. Combine the inequalities with "AND" since the region is between the two points.

    −2≤x≤4\boxed{-2 \le x \le 4}

    For the given points P1(3,2)P_1 (3, 2) and P2(−1,−2)P_2 (-1, -2).

    A) Find the slope (m).

  16. Identify the coordinates of the two points.

    (x1,y1)=(3,2)(x_1, y_1) = (3, 2) (x2,y2)=(−1,−2)(x_2, y_2) = (-1, -2)
  17. Use the slope formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.

    m=−2−2−1−3m = \frac{-2 - 2}{-1 - 3}
  18. Calculate the slope.

    m=−4−4m = \frac{-4}{-4} m=1\boxed{m = 1}

    B) By using the gradient - point Formula (m=y1−yx1−xm = \frac{y_1 - y}{x_1 - x}), formulate the function in the form y=mx+cy = mx + c.

  19. Substitute the calculated slope m=1m=1 and one of the points, for example P1(3,2)P_1(3, 2), into the given formula.

    1=2−y3−x1 = \frac{2 - y}{3 - x}
  20. Multiply both sides by (3−x)(3 - x).

    1(3−x)=2−y1(3 - x) = 2 - y 3−x=2−y3 - x = 2 - y
  21. Rearrange the equation to the form y=mx+cy = mx + c.

    y=2−(3−x)y = 2 - (3 - x) y=2−3+xy = 2 - 3 + x y=x−1\boxed{y = x - 1}

    C) What is the y - intercept?

  22. The y-intercept is the value of yy when x=0x = 0.

    Substitute x=0x=0 into the equation y=x−1y = x - 1.

    y=0−1y = 0 - 1
  23. Calculate the y-intercept.

    y=−1\boxed{y = -1}
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