SAT-Math-14: Graphing Linear Inequalities on the Coordinate Plane

1. The Two-Step Boundary and Shading Blueprint

Graphing a linear inequality builds directly on graphing a linear line, with two major visual twists: choosing the right type of line and shading the correct side.

Step 1: Draw the Boundary Line

Treat the inequality like a regular equation ($y = mx + b$) and sketch the line.

• Dashed Line (Shtrix chiziq): Use a dashed line if the symbol is $<$ or $>$. This means points exactly on the line are not part of the solution.

• Solid Line (Yaxlit chiziq): Use a solid line if the symbol is $\le$ or $\ge$. This means points on the line are part of the solution.

Step 2: Shade the Solution Region

An inequality represents an entire region of coordinate points, not just a single line.

• Shade ABOVE the line: If the equation is isolated as $y >$ or $y \ge$.

• Shade BELOW the line: If the equation is isolated as $y <$ or $y \le$.

🇺🇿 Uzbek Explanation:

Tengsizliklarni grafikda tasvirlash: Tengsizlik grafikda chiziq va bo'yalgan soha orqali ko'rsatiladi:

• Agar ishora $<$ yoki $>$ bo'lsa $\rightarrow$ chiziq shtrixli (dashed) bo'ladi.

• Agar ishora $\le$ yoki $\ge$ bo'lsa $\rightarrow$ chiziq yaxlit (solid) bo'ladi.

• $y >$ yoki $y \ge$ bo'lsa, chiziqdan tepa qism bo'yaladi.

• $y <$ yoki $y \le$ bo'lsa, chiziqdan pastki qism bo'yaladi.

2. The Foolproof Test Point Method

If an equation is written in a confusing layout (like standard form $Ax + By < C$) and you aren't sure which side to shade, use the Test Point Method.

1. Pick an easy coordinate point that is clearly not on the boundary line. The easiest point to test is almost always $(0,0)$.

2. Plug $x = 0$ and $y = 0$ into your inequality.

3. If the resulting statement is TRUE, shade the side of the line that contains $(0,0)$.

4. If the statement is FALSE, shade the opposite side away from $(0,0)$.

3. High-Yield SAT Practice Question

Question: Which of the following points $(x, y)$ is a solution to the inequality $2x - 3y < 6$?

A) $(4, 0)$

B) $(3, -1)$

C) $(1, 2)$

D) $(5, 1)$

Step-by-Step Explanation:

To find which point lives inside the shaded solution region without graphing, substitute each option directly into the inequality to find the true statement.

• Test A $(4, 0)$: $2(4) - 3(0) < 6 \implies 8 < 6$ (False)

• Test B $(3, -1)$: $2(3) - 3(-1) < 6 \implies 6 + 3 < 6 \implies 9 < 6$ (False)

• Test C $(1, 2)$: $2(1) - 3(2) < 6 \implies 2 - 6 < 6 \implies -4 < 6$ (True! $-4$ is definitely smaller than $6$.)

• Test D $(5, 1)$: $2(5) - 3(1) < 6 \implies 10 - 3 < 6 \implies 7 < 6$ (False)

Correct Answer: C

Summary

• Strict inequalities ($<, >$) use a dashed boundary; inclusive inequalities ($\le, \ge$) use a solid boundary.

• When $y$ is isolated, greater than means shade above, and less than means shade below.

• Use $(0,0)$ as a strategic test point to quickly verify which half-plane contains the correct solutions.

Fantastic job, my bro! Let's keep moving. Ready for SAT-Math-15: Modeling Real-World Scenarios with Inequalities?