Problem 16 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the inequality. $$5 x+3 y \geq-15$$

Step-by-Step Solution

Answer

The graph should include a solid line starting from the point (0, -5) and going down from left to right. The area above (and including) the line should be shaded.

Step 1 - Convert the Inequality to Equation

Convert the inequality into an equivalent equation by removing the inequality symbol. Here, the equation becomes \(5x + 3y = -15\).

Step 2 - Determine the Slope and Y-intercept

Write the equation in slope-intercept form. The slope-intercept form of a linear equation is \(y = mx + b\), where m is the slope of the line and b is the y-intercept. Rearrange the equation giving \(y=-\frac{5}{3}x -5\). Notice the slope, m, of the line is -5/3 and the y-intercept, b, is -5.

Step 3 - Drawing the Line on the Coordinate Plane

Start the line at the y-intercept which is the point (0, -5). Then plot another point by following the slope. Here, the slope is -5/3, it means we go down 5 units and move 3 units to the right. Draw a line through those points.

Step 4 - Apply the Inequality Sign

The inequality symbol, \(\geq\) means 'greater than or equal to'. As a result, the line drawn previously which corresponds to the equal case, should be a solid line. If it would have been a 'greater than' (>), we would have used a dashed line.

Step 5 - Shading the Area

Since our inequality is \(y \geq -\frac{5}{3}x -5\), we shade the area above the line (including the line itself because our inequality is 'greater than or equal to').