Question

from unit 4, lesson 4
here is a graph of the equation \\(6x + 2y = -8\\).
a. are the points \\((1.5, -4)\\) and \\((0, -4)\\) solutions to the equation?
explain or show how you know.

b. check if each of these points is a solution to the inequality \\(6x + 2y \le -8\\):
\\((-2, 2)\\)
\\((4, -2)\\)
\\((0, 0)\\)
\\((-4, -4)\\)

c. shade the solutions to the inequality.
d. are the points on the line included in the solution region?
explain how you know.

Explanation:

Verify points for the linear equation

Using the Constraint Modeling knowledge point

$$
LATEXBLOCK0
$$

Test points for the inequality

Using the Graphing Inequalities knowledge point

$$
LATEXBLOCK1
$$

Determine the shaded region

To shade the solutions to \(6x + 2y \le -8\), we test a point not on the line, such as \((0,0)\). Since \(0 \le -8\) is false, we shade the half-plane that does not contain \((0,0)\), which is the region below and to the left of the boundary line.

Analyze boundary line inclusion

The inequality operator is \(\le\) (less than or equal to). Because it includes "equal to," any coordinate pair lying directly on the boundary line \(6x + 2y = -8\) satisfies the inequality. Therefore, the points on the line are included in the solution region.

Answer:

Question a

• Are the points solutions?
• \((1.5, -4)\): No
• \((0, -4)\): Yes
• Explanation: Substituting \((1.5, -4)\) gives \(6(1.5) + 2(-4) = 1\), which does not equal \(-8\). Substituting \((0, -4)\) gives \(6(0) + 2(-4) = -8\), which is correct.

Question b

• \((-2, 2)\): Yes (solution)
• \((4, -2)\): No (not a solution)
• \((0, 0)\): No (not a solution)
• \((-4, -4)\): Yes (solution)

Question c

The solution region is shaded below and to the left of the boundary line \(6x + 2y = -8\).

Question d

• Are the points on the line included? Yes.
• Explanation: The inequality is non-strict (\(\le\)), meaning points where \(6x + 2y = -8\) are explicitly included in the solution set.