Problem 295 - Elementary Algebra

Question

In the following exercises, solve each system by graphing. $$ \left\\{\begin{array}{l} x-2 y<3 \\ y \leq 1 \end{array}\right. $$

Step-by-Step Solution

Answer

Graph the lines \(y = \frac{1}{2}x - \frac{3}{2}\) (dashed) and \(y = 1\) (solid), then find the overlap of the shaded regions above and below these lines, respectively.

Step 1: Write the inequalities in slope-intercept form

The inequalities are: 1. \(x - 2y < 3\) 2. \( y \leq 1 \) First, convert these inequalities to the slope-intercept form (\(y = mx + b\)). For the first inequality: \(x - 2y < 3\) implies \(-2y < -x + 3\), divide by -2 (and reverse the inequality sign) to get: \(y > \frac{1}{2}x - \frac{3}{2}\). The second inequality is already in slope-intercept form.

Step 2: Graph the boundary lines

Graph the boundary lines for each inequality on a coordinate plane.1. For \(y = \frac{1}{2}x - \frac{3}{2}\), sketch the line as a dashed line because the inequality is strict (\(y > \frac{1}{2}x - \frac{3}{2}\)). 2. For \(y = 1\), sketch the line as a solid line because the inequality includes equality (\(y \leq 1\)).

Step 3: Shade the appropriate regions

Shade the region above the dashed line (\(y > \frac{1}{2}x - \frac{3}{2}\)) since the inequality is 'greater than'. Shade the region below the solid line (\(y \leq 1\)) since the inequality is 'less than or equal to'.

Step 4: Identify the solution region

The solution region is the area where the shaded regions for both inequalities overlap. This represents all the points that satisfy both inequalities.

Key Concepts

Graphing Inequalities

Graphing inequalities involves plotting the boundary lines on a coordinate plane and shading the relevant regions. The boundary line of an inequality is derived by simply replacing the inequality symbol with an equals sign. For strict inequalities like < or >, this boundary is represented by a dashed line, indicating that points on the line itself are not included in the solution.
For inclusive inequalities like ≤ or ≥, a solid line is used, showing that points on the line are part of the solution. After plotting the boundary lines, the next step is to determine which side of the boundary line contains the solutions.
This is done by selecting a test point (commonly the origin, unless it lies on the boundary) and checking if it satisfies the inequality. If true, shade the region containing the test point; otherwise, shade the opposite side. Repeat this process for each inequality in your system. Lastly, the solution region is where the shaded areas overlap, representing all the points that fulfill all inequalities in the system.

Slope-Intercept Form

The slope-intercept form of a linear equation is written as:
\[ y = mx + b \], where:

• \textbf{m} is the slope, a measure of how steep the line is
  
• \textbf{b} is the y-intercept, the point where the line crosses the y-axis

Converting an inequality into slope-intercept form makes it easier to graph. For example, to convert the inequality \( x - 2y < 3 \) into slope-intercept form, solve for y:

• Rewrite the inequality: \( x - 2y < 3 \)
  
• Isolate y by subtracting x from both sides: \( -2y < -x + 3 \)
  
• Divide by -2 and reverse the inequality sign: \( y > \frac{1}{2}x - \frac{3}{2} \)
  

Now, this inequality is ready for graphing. The final inequality, \( y eq 1 \) is already in the slope-intercept form.

Solution Region

The solution region is the area on the graph where all inequalities overlap, representing the set of points that satisfy all conditions. To identify this region, follow these steps:

• Graph each inequality according to their respective conditions (dashed or solid lines).
  
• Shade the regions that satisfy each inequality.
  

For instance, in our example:

1. Graph \( y = \frac{1}{2}x - \frac{3}{2} \) with a dashed line since it is a 'greater than' inequality ( \( y > \frac{1}{2}x - \frac{3}{2} \)).
   
2. Shade the region above the dashed line because all points above this line satisfy the inequality.
   
3. Graph \( y = 1 \) with a solid line since it is a 'less than or equal to' inequality (\( y eq 1 \)).
   
4. Shade the region below the solid line because all points below satisfy this inequality.
   

The solution region is where the shaded areas from both inequalities overlap. This overlap represents all points that satisfy the system of inequalities. Always double-check by picking a point in the overlap region to confirm it fulfills all inequalities in the system.