Question

graph the inequality
$y \geq \frac{7}{6}x - 5$.

Explanation:

Step1: Identify the boundary line

The inequality is \( y \geq \frac{7}{6}x - 5 \). The boundary line is the equation \( y=\frac{7}{6}x - 5 \). Since the inequality is "greater than or equal to", the line will be solid (because the points on the line are included in the solution set).

Step2: Find two points on the boundary line

To graph the line \( y = \frac{7}{6}x-5 \), we can find the x - intercept and y - intercept or use other values of x.

• Y - intercept: When \( x = 0 \), \( y=\frac{7}{6}(0)-5=- 5 \). So one point is \( (0,-5) \).
• X - intercept: When \( y = 0 \), we solve the equation \( 0=\frac{7}{6}x-5 \).

Add 5 to both sides: \( 5=\frac{7}{6}x \).
Multiply both sides by \( \frac{6}{7} \): \( x=\frac{30}{7}\approx4.29 \). So another point is \( (\frac{30}{7},0) \) or approximately \( (4.29,0) \).
Plot the two points \( (0, - 5) \) and \( (\frac{30}{7},0) \) and draw a solid line through them.

Step3: Determine the region to shade

To determine which side of the line to shade, we can test a point not on the line. A common test point is the origin \( (0,0) \).
Substitute \( x = 0 \) and \( y = 0 \) into the inequality \( y\geq\frac{7}{6}x - 5 \):
\( 0\geq\frac{7}{6}(0)-5 \)
\( 0\geq - 5 \), which is true.
Since the test point \( (0,0) \) satisfies the inequality, we shade the region that includes the origin (the region above the line, because the inequality is \( y\geq\) the linear expression).

Answer:

1. Draw a solid line with the equation \( y=\frac{7}{6}x - 5 \) (using points like \( (0,-5) \) and \( (\frac{30}{7},0) \)).
2. Shade the region above the line (the region that includes the origin \( (0,0) \) as it satisfies \( y\geq\frac{7}{6}x - 5 \)).