Mathematics, 05.07.2019 12:30 RyanODON

Which graph represents the solution set of the system of inequalities? {x+y< 12 y≥x−4

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Answer from: lanaiheart7

Inequalities are used to relate unequal expressions.

See attachment for the graph that represents the solution set

The inequalities are given as:

$\mathbf{x + y < 12}$

$\mathbf{y \ge x - 4}$

Assume the inequalities are equations

$\mathbf{x + y = 12}$

$\mathbf{y = x - 4}$

Substitute $\mathbf{y = x - 4}$ in $\mathbf{x + y = 12}$

$\mathbf{x + x - 4 =12}$

$\mathbf{2x - 4 =12}$

Collect like terms

$\mathbf{2x = 4 +12}$

$\mathbf{2x = 16}$

Divide both sides by 2

$\mathbf{x = 8}$

Substitute $\mathbf{x = 8}$ in $\mathbf{y = x - 4}$

$\mathbf{y = 8 - 4}$

$\mathbf{y = 4}$

Back to the inequalities:

$\mathbf{x + y < 12}$

$\mathbf{y \ge x - 4}$

The solution to the inequalities would be:

$\mathbf{x < 8}$

$\mathbf{y \ge 4}$

$\mathbf{x < 8}$ and $\mathbf{x + y < 12}$ means that:

The line of the graph passes through x = 12 with a dotted lineThe left-hand side of the graph is shaded

$\mathbf{y \ge 4}$ and $\mathbf{y \ge x - 4}$ means that:

The line of the graph passes through y = -4 with a solid lineThe upper side of the graph is shaded

None of the given graphs represent the solution to the inequalities

See attachment for the graph that represents the solution set of the system of inequalities

Read more about inequalities at:

link

Which graph represents the solution set of the system of inequalities? {x+y< 12 y≥x−4

Answer from: lizzepacheco

The graph in the bottom right-hand corner

(see figure 4 in the attached images below)

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Explanation:

Let's start off by graphing x+y < 1. The boundary equation is x+y = 1 since we simply change the inequality sign to an equal sign. Solve for y to get x+y = 1 turning into y = -x+1. This line goes through (0,1) and (1,0). The boundary line is a dashed line due to the fact that there is no "or equal to" in the original inequality sign. So x+y < 1 turns into y < -x+1 and we shade below the dashed line. The "less than" means "shade below" when y is fully isolated like this. See figure 1 in the attached images below.

Let's graph 2y >= x-4. Start off by dividing everything by 2 to get y >= (1/2)x-2. The boundary line is y = (1/2)x-2 which goes through the two points (0,-2) and (4,0). The boundary line is solid. We shade above the boundary line. Check out figure 2 in the attached images below.

After we graph each individual inequality, we then combine the two regions on one graph. See figure 3 below. The red and blue shaded areas in figure 3 overlap to get the purple shaded area you see in figure 4, which is the final answer. Any point in this purple region will satisfy both inequalities at the same time. The solution point cannot be on the dashed line but it can be on the solid line as long as the solid line is bordering the shaded purple region. Figure 4 matches up perfectly with the bottom right corner in your answer choices.