Problem 285 - Elementary Algebra

Question

In the following exercises, solve each system by graphing. $$ \left\\{\begin{array}{l} y<2 x-1 \\ y \leq-\frac{1}{2} x+4 \end{array}\right. $$

Step-by-Step Solution

Answer

Graph the lines y=2x-1 (dashed) and y=-1/2x+4 (solid), shade the appropriate regions, and find the overlap.

Step 1 - Graph the first inequality

Start by graphing the boundary line for the inequality \( y = 2x - 1 \). This line has a slope (m) of 2 and a y-intercept (b) of -1. Because the inequality is y < 2x - 1, draw a dashed line (since the inequality is strict) and shade below the line.

Step 2 - Graph the second inequality

Next, graph the boundary line for the inequality \( y \leq -\frac{1}{2} x + 4 \). This line has a slope of -1/2 and a y-intercept of 4. Since the inequality is \( \leq \), draw a solid line and shade below the line.

Step 3 - Determine the solution region

The solution to the system is the region where the shaded areas from the two inequalities overlap. Identify this overlapping region on your graph.

Step 4 - Verify the solution

Choose a test point within the overlapping region (for example, the origin (0,0)) and substitute it into both inequalities to confirm it satisfies both conditions.

Key Concepts

Graphing Inequalities

Graphing inequalities is a key technique to visually solve systems of inequalities. When you graph an inequality, you begin by drawing its boundary line. The type of boundary line (dashed or solid) depends on whether the inequality is strict (< or >) or inclusive (≤ or ≥).
To graph the inequality, follow these steps:
1. Identify the slope and y-intercept from the equation of the boundary line.
2. Plot the y-intercept on the graph.
3. Use the slope to find at least one more point on the line.
4. Draw the boundary line (dashed for < or > and solid for ≤ or ≥).
Remember to always pay attention to the type of inequality as it determines the nature of the boundary line.

Slope and Y-Intercept

The slope and y-intercept are fundamental for graphing a line. The slope (m) represents the steepness or incline of a line and is calculated as the rise over the run (change in y divided by change in x).
The y-intercept (b) is the point where the line crosses the y-axis.
For example, in the inequality equation y < 2x - 1:
- The slope (m) is 2, meaning the line rises 2 units for every 1 unit it runs to the right.
- The y-intercept (b) is -1, meaning the line crosses the y-axis at (0, -1).
Accurately identifying and plotting the slope and y-intercept ensures you graph the line correctly, setting up the foundation to work with inequalities.

Shading Regions

Shading regions on a graph helps identify which side of the boundary line satisfies the inequality. After drawing the boundary line, determine which side of the line to shade:
- For y < (or y ≤), shade below the line.
- For y > (or y ≥), shade above the line.
Shading correctly is crucial as the solution to the inequality system is found in the intersection of these shaded regions. Check a test point not on the line (often the origin (0,0), if possible) to ensure the correct region is shaded. If the test point satisfies the inequality, shade the side where that point lies.

Boundary Lines

Boundary lines define the edges of the regions that satisfy an inequality. There are two types of boundary lines:
- Dashed lines indicate that points on the line do not satisfy the inequality (used for < or >).
- Solid lines indicate that points on the line do satisfy the inequality (used for ≤ or ≥).
In our example, the boundary line for y < 2x - 1 is dashed because the inequality is strict. The boundary line for y ≤ -1/2x + 4 is solid because the inequality includes equal to (≤).
Accurately drawing boundary lines ensures the correct representation of solutions and helps identify the intersection region where all conditions of the system are met.