Question

1. x + 3y < 15

Explanation:

Step1: Rewrite the inequality in slope - intercept form

We start with the inequality \(x + 3y<15\). To get it in the form \(y=mx + b\) (where \(m\) is the slope and \(b\) is the y - intercept), we solve for \(y\).
Subtract \(x\) from both sides: \(3y<-x + 15\).
Then divide each term by 3: \(y<-\frac{1}{3}x + 5\).

Step2: Graph the boundary line

The boundary line for the inequality \(y<-\frac{1}{3}x + 5\) is the line \(y =-\frac{1}{3}x+5\). Since the inequality is "less than" (not "less than or equal to"), we draw a dashed line.

• The y - intercept of the line \(y =-\frac{1}{3}x + 5\) is \(b = 5\), so the line passes through the point \((0,5)\).
• The slope \(m=-\frac{1}{3}\), which means for every 3 units we move to the right (increase in \(x\) by 3), we move down 1 unit (decrease in \(y\) by 1). So another point on the line can be found by starting at \((0,5)\), moving 3 units to the right to \(x = 3\) and 1 unit down to \(y=4\), so the point \((3,4)\) is also on the line.

Step3: Shade the region

To determine which side of the line to shade, we can use a test point. A common test point is the origin \((0,0)\) (as long as the line does not pass through it, which it doesn't in this case since when \(x = 0,y=5
eq0\)).
Substitute \(x = 0\) and \(y = 0\) into the original inequality \(x+3y<15\):
\(0+3(0)=0<15\), which is true.
So we shade the region that contains the origin \((0,0)\), which is the region below the dashed line \(y =-\frac{1}{3}x + 5\).

If you want to find the intercepts:

• For the x - intercept, set \(y = 0\) in the equation \(x+3y=15\) (the boundary line equation). Then \(x+3(0)=15\), so \(x = 15\). The x - intercept is \((15,0)\).
• For the y - intercept, set \(x = 0\) in the equation \(x + 3y=15\). Then \(0+3y=15\), so \(y = 5\). The y - intercept is \((0,5)\).

Answer:

To graph \(x + 3y<15\):

1. Rewrite as \(y<-\frac{1}{3}x + 5\).
2. Draw a dashed line \(y=-\frac{1}{3}x + 5\) (passing through \((0,5)\) and \((3,4)\) etc.).
3. Shade the region below the dashed line (the region containing \((0,0)\) as the test point \((0,0)\) satisfies \(x + 3y<15\)).