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Chapter 9: Problem 31

Graph the solution set of each system of inequalities. $$\begin{array}{r}2 x+y>2 \\\x-3 y<6\end{array}$$

Short Answer

Expert verified
Shade above the lines y = -2x + 2 and y = (1/3)x - 2. The intersecting region is the solution set.

Step by step solution

01

- Rewrite the inequalities in slope-intercept form

Rewrite each inequality so it has the form y = mx + b.For the first inequality: \[2x + y > 2\]Subtract 2x from both sides to get:\[y > -2x + 2\]For the second inequality:\[x - 3y < 6\]Subtract x from both sides to get:\[-3y < -x + 6\]Divide both sides by -3 (remember to reverse the inequality sign when dividing by a negative number):\[y > \frac{1}{3}x - 2\]
02

- Graph the boundary lines

Graph the boundary lines for each inequality on a coordinate plane.For \(y = -2x + 2\), draw a dashed line since the inequality is '>'.For \(y = \frac{1}{3}x - 2\), draw another dashed line since the inequality is also '>'.Remember, these lines represent the boundary and are not included in the solution set.
03

- Identify the regions to be shaded

Determine which side of each boundary line to shade.For the first inequality \(y > -2x + 2\), shade the region above the line because the '>' indicates that y is greater than the expression.For the second inequality \(y > \frac{1}{3}x - 2\), shade the area above this line as well.The solution set is the region where the shaded areas of both inequalities overlap.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

system of inequalities
A system of inequalities consists of multiple inequalities that you need to solve simultaneously. Each inequality represents a region on a coordinate plane. Finding the solution set for a system of inequalities means identifying the region where the solutions to all individual inequalities overlap.
Here, we are working with two inequalities:
  • (1) 2x + y > 2
  • (2) x - 3y < 6.
These inequalities will help us determine specific regions on the coordinate plane that satisfy both conditions.
slope-intercept form
The slope-intercept form is a way to express linear equations. It follows the format \(y = mx + b\). In this form,
  • 'm' represents the slope, showing how steep the line is.
  • 'b' represents the y-intercept, which is where the line crosses the y-axis.

    • To graph our inequalities, we first need to convert them to slope-intercept form:
    • Inequality (1): \(2x + y > 2\) becomes \(y > -2x + 2\) after moving x and simplifying.
    • Inequality (2): \(x - 3y < 6\) becomes \(y > \frac{1}{3}x - 2\) after moving x, division, and flipping the inequality sign.
boundary lines
Boundary lines are the lines we draw on the graph to represent the conditions of each inequality. They help us visualize which regions satisfy the inequalities.
For each 'slope-intercept' inequality, we will:
  • Draw the boundary line as if it were an equation. For example, the line for \(y = -2x + 2\).
  • Use a dashed line because the actual solution doesn't include the line itself due to the '>' in both inequalities.

    • We now have two dashed lines:
    • A line for \(y = -2x + 2\).
    • A line for \(y = \frac{1}{3}x - 2\).
shading regions
Shading regions helps us identify where our solutions overlap on the graph.
For each inequality, determine which side of the boundary line to shade:
  • For \(y > -2x + 2\), we shade the area above the line because y is greater.
  • For \(y > \frac{1}{3}x - 2\), we shade the area above this line as well.
The solution to our system of inequalities is the area where the shaded regions overlap. This is the set of points that satisfy both inequalities simultaneously.
To clearly see the overlap:
  • Carefully trace where the two shaded regions meet.

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