Problem 43 - Algebra 1

Question

Explain why the system of inequalities has no solution. $$\begin{array}{l} 2 x-y>4 \\ y>2 x-2 \end{array}$$

Step-by-Step Solution

Answer

The system of inequalities given has no solution because the areas defined by the inequalities do not overlap when graphed.

Step 1: Graph the first inequality

The first inequality \(2x - y > 4\) can be rewritten in slope-intercept form as \(y < 2x - 4\). This represents a line with a slope of 2 and a y-intercept of -4, from which y values are all less than the line. Visualize this graphically and make sure to shade the area beneath the line to represent the solutions for this inequality.

Step 2: Graph the second inequality

The second inequality \(y > 2x - 2\) is already in slope-intercept form. It represents a line with a slope of 2 and a y-intercept of -2. Here, the y values are greater than the line, so shade the area above the line to represent the solutions for this inequality.

Step 3: Find the overlapping region

Normally, for a system of inequalities to have a solution, the shaded areas for each inequality need to overlap, as the overlapping region would represent the solution for the whole system. However, in this particular case, no such overlapping can occur due to the distinct areas defined by the inequalities. Hence the system of inequalities has no solution.

Key Concepts

Graphing Inequalities

Graphing inequalities is a crucial skill when dealing with systems of inequalities. To graph an inequality, you first graph the corresponding equation as if the inequality sign were an equal sign. This gives you a line which represents all points that would be equal to a certain value. From there, the inequality sign tells you which side of the line contains the solutions. If the inequality is greater than (\textgreater) or less than (\textless), you shade above or below the line, respectively, to represent all the potential solutions. For example, with the inequality \(y < 2x - 4\), we graph the line \(y = 2x - 4\) and then shade below it because the 'less than' sign indicates that the valid solutions for \(y\) are lower than the points on the line.

Visualizing this on a coordinate plane helps you see where solutions exist, especially when comparing multiple inequalities. When dealing with a system, the solution is where the shaded areas overlap. However, if there is no overlap, as with the inequalities \(2x - y > 4\) and \(y > 2x - 2\), it means that there are no points that satisfy both conditions, thus resulting in no solution.

Slope-Intercept Form

The slope-intercept form of a line's equation is incredibly helpful for quickly graphing linear equations and inequalities. This form is written as \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept, the point where the line crosses the y-axis. In the context of the exercise, the inequalities were converted or identified in this form to facilitate graphing.

For the inequalities given, \(2x - y > 4\) can be manipulated into the slope-intercept form by subtracting \(2x\) from both sides and multiplying by -1 to obtain \(y < 2x - 4\). Notably, this form allows us to see that the line crosses the y-axis at -4 and rises with a slope of 2, indicating that for every one unit increase in \(x\), \(y\) increases by 2 units. Understanding and utilizing the slope-intercept form simplifies the process of graphing and finding solutions for inequalities.

Solutions for Inequalities

Determining solutions for inequalities involves finding all the values that make the inequality true. When dealing with a single inequality, any point on the graph that is in the shaded region represents a possible solution. In a system of inequalities, the solution is typically the set of points that are in the overlapping shaded regions from each individual inequality.

This exercise showcased a scenario where, after graphing both inequalities, there was no overlap between the shaded areas, indicating no shared solutions. This is a special case in systems of inequalities—it tells us that there are no possible values that would satisfy both inequalities simultaneously. When you encounter such a situation, you can confidently state that the system of inequalities has no solution. Understanding this concept is crucial for accurately interpreting and solving inequality-based problems.

Linear Equations

Linear equations form the foundation of the lines we graph when working with inequalities. They are equations in which variables appear to the first power only (not squared, cubed, etc.), and they graph as straight lines. The standard form of a linear equation is \(Ax + By = C\), but for graphing, it is often beneficial to rearrange them into slope-intercept form as described earlier.

In this exercise, the linear equations we derived from the inequalities create two parallel lines, which graphically represent all the points that meet either the first or second equation. Since parallel lines never intersect, there cannot be a common solution—a fact that is visually evident because the shaded regions don't overlap. Recognizing linear equations and their graphical representations is an essential component in understanding why some systems of inequalities may not have solutions, as seen in the provided problem.