Question

solve the inequality and then graph your solution. (chapter 2)
$4x + 6y \geq 10$

a.

c.

b.

d.

\bigcirc graph a
\bigcirc graph d

Explanation:

Step1: Rewrite the inequality

Start with the inequality \(4x + 6y \geq 10\). Solve for \(y\) to get it in slope - intercept form (\(y=mx + b\)).
Subtract \(4x\) from both sides: \(6y\geq - 4x + 10\).
Divide each term by 6: \(y\geq-\frac{4}{6}x+\frac{10}{6}\), simplify to \(y\geq-\frac{2}{3}x+\frac{5}{3}\).

Step2: Analyze the boundary line

The slope of the line \(y =-\frac{2}{3}x+\frac{5}{3}\) is \(m =-\frac{2}{3}\) (negative slope) and the y - intercept \(b=\frac{5}{3}\approx1.67\). Since the inequality is \(\geq\), the boundary line should be solid (not dashed).

Step3: Test a point

Let's test the point \((0,0)\) in the original inequality \(4x + 6y\geq10\). Substitute \(x = 0\) and \(y = 0\): \(4(0)+6(0)=0\), and \(0\geq10\) is false. So the region that satisfies the inequality should not include the origin.

Step4: Analyze the graphs

• For a line with a negative slope (\(m =-\frac{2}{3}\)) and a solid line (because of \(\geq\)) and the region not including the origin:
• Graphs with dashed lines (b) can be eliminated.
• Graphs that include the origin (need to check the shading) can be eliminated. The line \(y =-\frac{2}{3}x+\frac{5}{3}\) has a positive y - intercept. Let's check the y - intercept of the lines in the graphs. The line in graph c has a y - intercept that is positive (around 2, close to \(\frac{5}{3}\approx1.67\)) and the slope is negative. Also, the shading in graph c does not include the origin (since when we test \((0,0)\), it does not satisfy the inequality and the shading is on the side that does not include \((0,0)\)) and the line is solid.

Answer:

The correct graph is Graph C (assuming Graph C is the one with a solid line, negative slope, positive y - intercept, and shading not including the origin). If we consider the labels, if Graph C is the one in the upper - right with the solid line and appropriate shading, then the answer is the graph corresponding to the inequality \(y\geq-\frac{2}{3}x+\frac{5}{3}\), which should be the graph with a solid line, negative slope, y - intercept around \(\frac{5}{3}\), and shading above the line (since \(y\geq\) the line) and not including the origin.