Graphing Linear Inequalities

Step-by-Step Process

1. Graph the boundary line (treat the inequality as an equation)

   • Use a solid line for $\leq$ or $\geq$ (boundary IS included)
   • Use a dashed line for $<$ or $>$ (boundary is NOT included)

2. Choose a test point not on the line ($(0, 0)$ is easiest if the line doesn't pass through it)

3. Shade the correct side

   • If the test point satisfies the inequality → shade that side
   • If not → shade the other side

Reading Inequality Graphs

Systems of Inequalities

A system of inequalities is two or more inequalities graphed on the same coordinate plane. The solution is the overlapping shaded region.

How to Find the Solution Region

1. Graph each inequality separately
2. The solution is where ALL shadings overlap
3. Any point in the overlapping region satisfies ALL inequalities

Interpreting Constraints (SAT Favorite)

The SAT loves to give real-world constraint problems:

Example: A bakery makes cookies ($c$) and brownies ($b$).

• Each cookie uses 2 eggs; each brownie uses 3 eggs
• They have at most 120 eggs
• They must make at least 10 cookies and at least 5 brownies

$2c + 3b \leq 120$ $c \geq 10$ $b \geq 5$

The feasible region is where all three inequalities overlap.

SAT Question Types

Type 1: "Which inequality represents the graph?"

Look at:

• Slope and $y$-intercept of the boundary line
• Whether the line is solid or dashed
• Which side is shaded

Type 2: "Which point is in the solution region?"

Test each answer choice in ALL inequalities. The point must satisfy every inequality.

Type 3: "What is the maximum/minimum value?"

Check the vertices (corner points) of the feasible region. The max/min of a linear expression always occurs at a vertex.

Type 4: "Set up the inequality"

Translate a word problem into an inequality using key phrases like "at most" ($\leq$), "at least" ($\geq$), "fewer than" ($<$), "more than" ($>$).

Compound Inequalities on a Number Line

A compound inequality combines two inequalities:

AND compound: $-3 < x \leq 5$ means $x$ is between $-3$ and $5$ (inclusive of 5)

• Graph: open circle at $-3$, closed circle at $5$, shade between

OR compound: $x < -2$ or $x > 4$ means $x$ is outside $(-2, 4)$

• Graph: shade left of $-2$ and right of $4$

Common SAT Mistakes

1. Using the wrong line type: Solid vs. dashed matters!
2. Shading the wrong side: Always test a point
3. Forgetting to flip the inequality when multiplying/dividing by a negative
4. Confusing AND vs. OR in compound inequalities
5. Not checking ALL inequalities when testing a point in a system

Quick Strategy Guide

• "At most" = $\leq$ (can be equal to or less than)
• "At least" = $\geq$ (can be equal to or greater than)
• "No more than" = $\leq$
• "No fewer than" = $\geq$
• When in doubt, test the point $(0, 0)$ to determine which side to shade

Step 3: Verify $(2, 4)$ satisfies both inequalities:

• $y \leq -x + 6$: $4 \leq -2 + 6 = 4$ ✓ (equal, so on boundary)
• $y \geq 2x$: $4 \geq 2(2) = 4$ ✓ (equal, so on boundary)

Answer: $(2, 4)$

Strategy: "On the boundary of both" means find the intersection of the boundary lines.

Step 3: Find the vertices of the feasible region.

Intersection of $3a + 2b = 240$ and $2a + 4b = 200$: Multiply first equation by 2: $6a + 4b = 480$ Subtract: $6a + 4b - 2a - 4b = 480 - 200 → 4a = 280 → a = 70$ $2(70) + 4b = 200 → 4b = 60 → b = 15$ Vertex: $(70, 15)$

Other vertices:

• $(0, 0)$: Profit = $0
• $(80, 0)$: From $3a = 240$, $a = 80$. Check material: $2(80) = 160 \leq 200$ ✓
• $(0, 50)$: From $4b = 200$, $b = 50$. Check labor: $2(50) = 100 \leq 240$ ✓

Step 4: Evaluate profit $P = 50a + 40b$ at each vertex:

• $(0, 0)$: $P = 0$
• $(80, 0)$: P = 50(80) = \4,000$
• $(0, 50)$: P = 40(50) = \2,000$
• $(70, 15)$: P = 50(70) + 40(15) = 3,500 + 600 = \4,100$

Answer: Maximum profit = $4,100 at $(70, 15)$