Write A System Of Linear Inequalities Represented By The Graph

Understanding how to write a system of linear inequalities represented by a graph is a fundamental skill in algebra that bridges visual interpretation with algebraic notation. This process requires analyzing boundary lines, determining slope and intercepts, identifying shading regions, and translating these visual cues into precise mathematical symbols. Whether you are a student preparing for an exam or an educator looking for a structured way to explain the concept, mastering this translation builds a strong foundation for linear programming and optimization problems later on.

Short version: it depends. Long version — keep reading.

Understanding the Core Components

Before diving into the step-by-step process, it is essential to recognize the three critical elements present in every graph of a linear inequality system: the boundary lines, the shading (solution region), and the line types (solid vs. Because of that, dashed). Each element carries specific algebraic meaning.

Boundary Lines are the visual representation of the related linear equations (e.g., $y = mx + b$ or $Ax + By = C$). They act as the "borders" of the solution set.

Line Type dictates the inequality symbol:

• A solid line indicates that points on the line are included in the solution. This corresponds to $\le$ (less than or equal to) or $\ge$ (greater than or equal to).
• A dashed (or dotted) line indicates that points on the line are excluded from the solution. This corresponds to ${content}lt;$ (less than) or ${content}gt;$ (greater than).

Shading represents the half-plane containing all coordinate pairs $(x, y)$ that satisfy the inequality. The overlap (intersection) of all shaded regions represents the solution set for the entire system.

Step-by-Step Guide to Writing the System

Follow this structured workflow to accurately convert a graph into a system of linear inequalities.

1. Identify Each Boundary Line Individually

Treat every line on the graph as a separate entity. For a system with two inequalities, you will analyze two lines; for three inequalities, three lines, and so on.

2. Determine the Equation of the Boundary Line

Find the equation of the line in slope-intercept form ($y = mx + b$) or standard form ($Ax + By = C$). Slope-intercept form is often easier for determining the inequality direction later Turns out it matters..

• Find the y-intercept ($b$): Look where the line crosses the y-axis.
• Calculate the slope ($m$): Use the "rise over run" method. Select two clear lattice points (integer coordinates) on the line. $m = \frac{y_2 - y_1}{x_2 - x_1}$.
• Write the equation: Substitute $m$ and $b$ into $y = mx + b$.

Example: If a line crosses the y-axis at $(0, 3)$ and passes through $(2, 0)$, the slope is $\frac{0-3}{2-0} = -\frac{3}{2}$. The equation is $y = -\frac{3}{2}x + 3$.

3. Analyze the Line Style (Solid vs. Dashed)

Inspect the line style for the specific boundary you are analyzing.

• Solid Line $\rightarrow$ Use $\le$ or $\ge$.
• Dashed Line $\rightarrow$ Use ${content}lt;$ or ${content}gt;$.

4. Determine the Inequality Direction (The "Test Point" Method)

This is the most common source of errors. You must decide if the shading is above the line ($y$ is greater) or below the line ($y$ is less) It's one of those things that adds up..

The Standard Test Point Method:

1. Pick a point not on the line. The origin $(0,0)$ is the easiest choice, provided the line does not pass through it.
2. Substitute the coordinates of the test point into the equation of the line (treating it temporarily as an equality).
3. Check if the statement is True or False.
4. If True: The test point lies in the shaded region. The inequality symbol points toward the test point.
5. If False: The test point lies in the unshaded region. The inequality symbol points away from the test point.

The "Slope-Intercept Shortcut" (Only valid for $y = mx + b$ form):

• If the inequality is solved for $y$ (e.g., $y < mx + b$ or $y \ge mx + b$):
  • ${content}lt;$ or $\le$ $\rightarrow$ Shade BELOW the line.
  • ${content}gt;$ or $\ge$ $\rightarrow$ Shade ABOVE the line.
• Warning: This shortcut fails if the inequality is in standard form ($Ax + By < C$) or if you multiplied/divided by a negative number to isolate $y$. Always verify with a test point if unsure.

5. Write the Final Inequality

Combine the equation from Step 2 with the symbol from Steps 3 and 4 But it adds up..

6. Repeat for All Lines

Perform Steps 1–5 for every boundary line shown in the graph.

7. Compile the System

Write the inequalities together using a curly brace ${ }$ or simply list them separated by commas/line breaks, typically labeled as a "System of Linear Inequalities."

Detailed Worked Example

Let’s apply this workflow to a hypothetical graph containing two lines.

Graph Description:

• Line A: Solid line passing through $(0, 2)$ and $(3, 0)$. The region below the line (toward the origin) is shaded.
• Line B: Dashed vertical line at $x = -1$. The region to the right of the line is shaded.

Analyzing Line A

1. Equation:
   • y-intercept $b = 2$.
   • Slope $m = \frac{0 - 2}{3 - 0} = -\frac{2}{3}$.
   • Equation: $y = -\frac{2}{3}x + 2$.
2. Line Style: Solid $\rightarrow$ $\le$ or $\ge$.
3. Test Point $(0,0)$:
   • Substitute into equation: $0 \stackrel{?}{=} -\frac{2}{3}(0) + 2 \rightarrow 0 \stackrel{?}{=} 2$.
   • Statement: False ($0$ is not equal to $2$).
   • Wait, we check the inequality direction. Is $(0,0)$ in the shaded region? Yes (graph says region below line toward origin is shaded).
   • Since $(0,0)$ makes the equation false but lies in the shaded solution region, we need an inequality where $(0,0)$ makes it True.
   • Test $y \le -\frac{2}{3}x + 2$: $0 \le 2$ $\rightarrow$ True.
   • Test $y \ge -\frac{2}{3}x + 2$: $0 \ge 2$ $\rightarrow$ False.
   • Correct Symbol: $\le$.
4. Inequality A: $y \le -\frac{2}{3}x + 2$.

Analyzing Line B

1. Equation: Vertical line crossing x-axis at $-1$. Equation: $x = -1$. (Slope is undefined; cannot use $y=mx+b$).
2. Line Style: Dashed $\rightarrow$ ${content}lt;$ or ${content}gt;$.
3. Test Point $(0,0)$:
   • Is $(0,0)$ shaded? Yes (region to the right is shaded).
   • We need an inequality where $x=0$ is True.
   • Test $x < -1$: $0 < -1$ $\rightarrow$ False.
   • Test $x > -1$: $