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4-65.

Find boundary points for each of the following inequalities. Draw the boundaries on a number line and shade the solution regions. Homework Help ✎

  1. Solve for x to find the boundary point(s).

    3x + 2 = x − 6
    2x = −8
    x = −4

    Since x = −4, plot −4 on a number line.
    Since −4 makes the inequality true, shade the point.

    Test a point on either side of the boundary point in the original inequality and decide if you get a true or false statement.

    If you choose x = 0:
    3(0) + 2 ≥ (0) − 6
    2 ≥ −6 → True

    If you choose x = −6:
    3(−6) + 2 ≥ (−6) − 6
    −16 ≥ −12 → False

    Shade the points that are on the true side, to the right of −4.

  1. Solve for x to find the boundary point(s).

    2x2 − 5x = 12
    2x2 − 5x − 12 = 0

    Plot the boundary points on a number line. Since the points do not make the inequality true, do not shade them.

    Test a point between the boundary points in the original inequality. Decide if the simplified statement is true or false.

    If you choose x = 0:
    2(0)2 − 5(0) < 12
    0 < 12 → True

    Shade the points between the boundary points, the true region.