Question

1. 6x > 2y - 10

check:

Explanation:

Step1: Rearrange the inequality

We start with the inequality \(6x > 2y - 10\). First, we want to solve for \(y\) to get it in slope - intercept form (\(y=mx + b\)) which is easier to graph.
Add 10 to both sides of the inequality: \(6x + 10>2y\).
Then, divide both sides by 2: \(\frac{6x + 10}{2}>y\), which simplifies to \(3x+5 > y\) or \(y < 3x + 5\).

Step2: Identify the boundary line

The boundary line for the inequality \(y < 3x + 5\) is the line \(y=3x + 5\). Since the inequality is strict (\(<\), not \(\leq\)), the boundary line will be a dashed line.
To find two points on the line \(y = 3x+5\):

• When \(x = 0\), \(y=3(0)+5=5\). So the point \((0,5)\) is on the line.
• When \(x=-1\), \(y=3(-1)+5 = 2\). So the point \((-1,2)\) is on the line.

Step3: Determine the region to shade

We test a point not on the line, usually the origin \((0,0)\) (since it is not on \(y = 3x+5\) as \(0
eq3(0)+5 = 5\)).
Substitute \(x = 0\) and \(y = 0\) into the inequality \(y<3x + 5\): \(0<3(0)+5\), which simplifies to \(0 < 5\). This is a true statement. So we shade the region that contains the origin \((0,0)\).

Step4: Graph the inequality

1. Draw the dashed line through the points \((0,5)\) and \((-1,2)\) (using the slope \(m = 3\) and y - intercept \(b = 5\)).
2. Shade the region below the dashed line (since \(y<3x + 5\)) which includes the origin.

Answer:

The graph of the inequality \(6x>2y - 10\) (or \(y < 3x + 5\)) has a dashed line with slope 3 and y - intercept 5, and the region below the line (containing the origin) is shaded.