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Solve the following inequalities: (a) 3x ≤ 2x + 7 (b) 3x ≥ 10 (c) -x - 4 ≥ 3x (d) 7 ≥ 7 - 2x 2. Show the...

1. Solve the following inequalities:
(a) 3x ≤ 2x + 7
(b) 3x ≥ 10
(c) -x - 4 ≥ 3x
(d) 7 ≥ 7 - 2x
2. Show the region x + y ≤ 6 on a graph.
3. Show by shading the unwanted regions, the region satisfying these inequalities y ≥ x, x ≥ 3, and y ≤ 3 on the same graph.
4. Show by shading the unwanted regions, the region satisfying these inequalities x ≥ y, y ≤ 7, x ≤ 5, and 3x + 5y ≤ 30 on the same graph.

Answer

  1. Solve the following inequalities:

    1. 3x ≤ 2x + 7
      Subtract 2x from both sides:
      x ≤ 7

    2. 3x ≥ 10
      Divide both sides by 3:
      x ≥ 10/3

    3. -x - 4 ≥ 3x
      Add x to both sides:
      -4 ≥ 4x
      Divide by 4:
      -1 ≥ x

    4. 7 ≥ 7 - 2x
      Subtract 7 from both sides:
      0 ≥ -2x
      Divide by -2 (reverse inequality):
      0 ≤ x

  2. Show the region x + y ≤ 6 on a graph:
    Graph the line x + y = 6 and shade the region below it.

  3. Show by shading the unwanted regions, the region satisfying these inequalities y ≥ x, x ≥ 3, and y ≤ 3 on the same graph:
    Graph the lines y = x, x = 3, and y = 3. Shade the region where y ≥ x, x ≥ 3, and y ≤ 3 intersect.

  4. Show by shading the unwanted regions, the region satisfying these inequalities x ≥ y, y ≤ 7, x ≤ 5, and 3x + 5y ≤ 30 on the same graph:
    Graph the lines x = y, y = 7, x = 5, and 3x + 5y = 30. Shade the region where x ≥ y, y ≤ 7, x ≤ 5, and 3x + 5y ≤ 30 intersect.

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