Chapter 5: Problem 32
Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{l}5 x-3 y>-6 \\ 5 x-3 y<-9\end{array}\right.$$
Short Answer
Expert verified
The solution of the system of inequalities is the overlapping shaded areas above the line \(y = \frac{5}{3}x - 3\) and below the line \(y = \frac{5}{3}x + 2\).
Step by step solution
01
Rewrite the inequalities in slope-intercept form
The inequality equations are given as linear expressions. Before we can graph them, it's important to change them into the slope-intercept form \(y = mx + b\), where m is the slope and b is the y-intercept. So, rewriting our two inequalities we have \(-3y > -5x - 6\) and \(-3y < -5x + 9\). Dividing all terms by -3 and remembering that when we multiply or divide an inequality by a negative number, the direction of the inequality changes, we get \(y < \frac{5}{3}x + 2\) and \(y > \frac{5}{3}x - 3\) respectively.
02
Graph the inequalities
Now, we graph the two inequalities. This is done by firstly graphing the lines \(y = \frac{5}{3}x + 2\) and \(y = \frac{5}{3}x - 3\). These will be solid lines because the original inequalities do not include equality. Then, since the first inequality is \(y < \frac{5}{3}x + 2\), the area below this line is shaded and since the second inequality is \(y > \frac{5}{3}x - 3\), the area above this line is shaded. The overlapping shaded areas represent the solution of the system of inequalities.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Inequalities
Graphing inequalities involves visually representing the solutions to inequality equations on a coordinate plane. When dealing with a system of inequalities, you'll need to graph each inequality individually and then identify the overlapping region that satisfies all of them.
Start by converting the inequalities into linear equations by temporarily replacing inequality symbols with an equals sign. This produces boundary lines which you can graph. These lines divide the plane into two halves—one where the inequality holds true, and one where it does not.
Start by converting the inequalities into linear equations by temporarily replacing inequality symbols with an equals sign. This produces boundary lines which you can graph. These lines divide the plane into two halves—one where the inequality holds true, and one where it does not.
- Use a solid line for inequalities that include equality (≥ or ≤).
- Use a dashed line for inequalities that do not include equality (> or <).
- Check each inequality by testing a point not on the line (often (0,0) is used if it's not on any line), to see which side of the line to shade.
- The final solution is where all shaded regions overlap.
Slope-Intercept Form
The slope-intercept form is a way of writing linear equations. It is expressed as \( y = mx + b \). In this form:
When converting an inequality, rearrange the equation to isolate \(y\) on one side. Don’t forget if you multiply or divide by a negative number, the inequality flips direction!
For example, transforming the inequality \(-3y > -5x - 6\) to slope-intercept form gives you \(y < \frac{5}{3}x + 2\). Once in this form, graphing becomes straightforward as you follow the easy pattern of slope and intercept.
- \(m\) represents the slope of the line.
- \(b\) indicates the y-intercept—where the line crosses the y-axis.
When converting an inequality, rearrange the equation to isolate \(y\) on one side. Don’t forget if you multiply or divide by a negative number, the inequality flips direction!
For example, transforming the inequality \(-3y > -5x - 6\) to slope-intercept form gives you \(y < \frac{5}{3}x + 2\). Once in this form, graphing becomes straightforward as you follow the easy pattern of slope and intercept.
Solution Set
The solution set of a system of inequalities is the area on the graph where the solutions to all inequalities overlap. This represents all the pairs of \((x, y)\) that satisfy both inequalities in the system at once.
To find the solution set, graph each inequality. Look for the regions that are shaded multiple times, indicating that they satisfy all inequalities involved.
To find the solution set, graph each inequality. Look for the regions that are shaded multiple times, indicating that they satisfy all inequalities involved.
- The region should be bounded by the graphed lines.
- If the lines are dashed or solid depends on the original inequality signs.
Linear Equations
Linear equations are equations that form a straight line when graphed on a coordinate plane. They can be written in different forms, with the slope-intercept form being one of the most convenient for graphing.
Characteristics of linear equations include:
In systems of inequalities like our example, the linear equations define the boundary lines above or below which certain conditions (dictated by the inequality signs) hold true. Graphing these lines is an essential skill when visually solving systems of inequalities.
Characteristics of linear equations include:
- They have no exponents higher than 1 for the variables involved.
- They produce straight lines on a graph, depicting a constant rate of change.
In systems of inequalities like our example, the linear equations define the boundary lines above or below which certain conditions (dictated by the inequality signs) hold true. Graphing these lines is an essential skill when visually solving systems of inequalities.