Problem 32 - College Algebra

Question

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} y<-2 x+4 \\ y

Step-by-Step Solution

Answer

The solution of the system of inequalities consists of the area that is below both of the lines represented by y<-2x+4 and y

Step 1: Graph the first inequality

The first inequality is \(y < -2x + 4\). This is in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Here, the slope \(m\) is -2 and the y-intercept \(b\) is 4. Start by plotting the y-intercept on the y-axis. From there, use the slope to find another point. Since the slope is -2, this means 'down 2 units, right 1 unit'. So start from y-intercept (0,4) and make a line pointing below because y should be less than -2x+4.

Step 2: Graph the second inequality

The second inequality is \(y < x - 4\). The slope \(m\) is 1 and the y-intercept \(b\) is -4. Just as step 1, plot the y-intercept on the y-axis and from there, use the slope to find another point. Since the slope is 1, this means 'up 1 unit, right 1 unit'. So start from y-intercept (0,-4) and draw a line pointing below as y should be less than x-4.

Step 3: Identify the intersection of the two areas

Since both inequalities contain less than symbol, the lines drawn for both equations will be dashed lines. The area of the solution for this system of inequalities is the overlapping region that satisfies both inequalities. So the area which lies below both lines is the solution to the system.

Key Concepts

Solution Set of Inequalities

Understanding the solution set of inequalities is key when dealing with systems of inequalities in algebra. The solution set refers to all the possible values that satisfy all the inequalities in the system. To visualize this, imagine each inequality as a separate condition that carves out a region on the coordinate plane. When we deal with two or more inequalities, we're looking for the area where these regions overlap.

Each region is bounded by the line related to its inequality, but here's the catch—the boundary may or may not be part of the solution set. For example, if the inequality is strict (like < or >), we use a dashed line to denote the boundary is not included in the solution. On the other hand, if the inequality includes equality (like ≤ or ≥), it's represented by a solid line, indicating the boundary is part of the solution.

The solution set for a system is where you can shade on the graph indicating all points that satisfy every single inequality involved. This overlap is key—it's the 'magic zone' where all conditions are happy and live in harmony.

Inequality Graphing

The art of inequality graphing lays the foundation for understanding complex algebraic concepts. Graphing an inequality is similar to graphing a straight line, but with the added twist of a region that represents all the solutions to the inequality. To correctly graph an inequality, you'll need to start by identifying its boundary line—this is the line you would get if the inequality were an equation.

After plotting the boundary line (using a dashed or solid line as appropriate), the next step is to determine which side of the line is part of the solution set. A practical way is to pick a test point not on the line (often the origin is convenient, unless it's on the line) and substitute its coordinates into the inequality. If the inequality holds true, then the side containing the test point is shaded; if not, the opposite side is shaded.

By doing this for all inequalities in the system, you create a graphical representation where the regions of the different inequalities intersect. The shared shaded area represents the set of solutions that satisfies all inequalities simultaneously.

Slope-Intercept Form

The slope-intercept form of a line, expressed as y = mx + b, is a super highway to graphing lines quickly. In this formula, m stands for the slope, which tells us how steep the line is. The slope is calculated as 'rise over run.' This means for every unit the line goes up or down (rise), it goes across (run) a corresponding amount.

The y-intercept, represented by b, is where the line crosses the y-axis. That's our starting point when graphing. To sketch the graph of a line, you plot the y-intercept and then use the slope to find another point. For instance, if the slope is 2, you move up 2 units (rise) for every 1 unit you move to the right (run).

In the context of inequalities, these lines become the boundaries of your shaded regions. Remember, the line itself is not always part of the solution—it depends on whether the inequality symbol is strict or inclusive. Recognizing slope-intercept form quickly sets you up for faster graphing and better understanding of the solutions.

Previous

Problem 32

Next

Problem 32

Other exercises in this chapter

Problem 31

Solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution s

View solution

Problem 32

Write the partial fraction decomposition of each rational expression. $$\frac{9 x+2}{(x-2)\left(x^{2}+2 x+2\right)}$$

View solution

Problem 32

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} x^{2}+4 y^{2}-20 \\ x y-4 \end{array}\right.$$

View solution

Problem 32

Solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution s

View solution

Key Concepts

Solution Set of Inequalities

Understanding the solution set of inequalities is key when dealing with systems of inequalities in algebra. The solution set refers to all the possible values that satisfy all the inequalities in the system. To visualize this, imagine each inequality as a separate condition that carves out a region on the coordinate plane. When we deal with two or more inequalities, we're looking for the area where these regions overlap.

Each region is bounded by the line related to its inequality, but here's the catch—the boundary may or may not be part of the solution set. For example, if the inequality is strict (like < or >), we use a dashed line to denote the boundary is not included in the solution. On the other hand, if the inequality includes equality (like ≤ or ≥), it's represented by a solid line, indicating the boundary is part of the solution.

The solution set for a system is where you can shade on the graph indicating all points that satisfy every single inequality involved. This overlap is key—it's the 'magic zone' where all conditions are happy and live in harmony.

Inequality Graphing

The art of inequality graphing lays the foundation for understanding complex algebraic concepts. Graphing an inequality is similar to graphing a straight line, but with the added twist of a region that represents all the solutions to the inequality. To correctly graph an inequality, you'll need to start by identifying its boundary line—this is the line you would get if the inequality were an equation.

After plotting the boundary line (using a dashed or solid line as appropriate), the next step is to determine which side of the line is part of the solution set. A practical way is to pick a test point not on the line (often the origin is convenient, unless it's on the line) and substitute its coordinates into the inequality. If the inequality holds true, then the side containing the test point is shaded; if not, the opposite side is shaded.

By doing this for all inequalities in the system, you create a graphical representation where the regions of the different inequalities intersect. The shared shaded area represents the set of solutions that satisfies all inequalities simultaneously.

Slope-Intercept Form

The slope-intercept form of a line, expressed as y = mx + b, is a super highway to graphing lines quickly. In this formula, m stands for the slope, which tells us how steep the line is. The slope is calculated as 'rise over run.' This means for every unit the line goes up or down (rise), it goes across (run) a corresponding amount.

The y-intercept, represented by b, is where the line crosses the y-axis. That's our starting point when graphing. To sketch the graph of a line, you plot the y-intercept and then use the slope to find another point. For instance, if the slope is 2, you move up 2 units (rise) for every 1 unit you move to the right (run).

In the context of inequalities, these lines become the boundaries of your shaded regions. Remember, the line itself is not always part of the solution—it depends on whether the inequality symbol is strict or inclusive. Recognizing slope-intercept form quickly sets you up for faster graphing and better understanding of the solutions.