Question

questions
solve the following system of inequalities graphically on the set of axes below. state the coordinates of a point in the solution set.
\\( y < -x + 1 \\)
\\( y < \frac{1}{5}x + 7 \\)
answer
point ( , )

Explanation:

Step 1: Analyze the first inequality \( y < -x + 1 \)

The boundary line for \( y = -x + 1 \) has a slope of \(-1\) and a y - intercept of \(1\). Since the inequality is \( y < -x + 1 \), we draw a dashed line (because the inequality is strict, \(y\) is not equal to \(-x + 1\)) and shade the region below the line.

Step 2: Analyze the second inequality \( y < \frac{1}{5}x + 7 \)

The boundary line for \( y=\frac{1}{5}x + 7\) has a slope of \(\frac{1}{5}\) and a y - intercept of \(7\). Since the inequality is \(y < \frac{1}{5}x+7\), we draw a dashed line and shade the region below the line.

Step 3: Find the intersection region

The solution set of the system of inequalities is the region that is shaded for both inequalities. To find a point in the solution set, we can choose a point that satisfies both inequalities. Let's test the point \((0,0)\):

For the first inequality: \(0 < - 0+1\), which simplifies to \(0 < 1\), this is true.

For the second inequality: \(0<\frac{1}{5}(0)+7\), which simplifies to \(0 < 7\), this is also true.

Answer:

The point \((0,0)\) is in the solution set. (Other valid points could also be chosen, for example, \((- 5,0)\) : For \(y < -x + 1\), \(0 < -(-5)+1=6\) (true); for \(y < \frac{1}{5}x + 7\), \(0<\frac{1}{5}(-5)+7=- 1 + 7 = 6\) (true))