### Home > CCA2 > Chapter 4 > Lesson 4.2.1 > Problem4-65

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. $3x + 2 ≥ x − 6$

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. $2x^2 − 5x < 12$

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.