SAT — mathHard7 min read

Systems of Linear Inequalities

Master systems of linear inequalities for the Digital SAT. Find feasible regions, identify solution points, and interpret constraints.

Tutor AI Team2026-02-25T01:17:12.160Z

A system of linear inequalities is a set of two or more inequalities that must be satisfied simultaneously. The solution is a region in the coordinate plane where the shaded areas of all inequalities overlap — called the feasible region. The Digital SAT tests this concept through graphs, word problems involving constraints, and questions about which points lie in the solution region.

Core Concepts

Graphing a System of Inequalities

Graph each inequality separately (boundary line + shading).
The solution region is where all shading overlaps.

Example: $y \leq 2x + 3$ $y > -x + 1$

Graph $y = 2x + 3$ as a solid line, shade below.
Graph $y = -x + 1$ as a dashed line, shade above.
The solution is the overlapping region.

Identifying Points in the Feasible Region

A point $(a, b)$ is a solution to the system if it satisfies every inequality.

Example: Is $(2, 5)$ a solution to $y \leq 2x + 3$ and $y > -x + 1$ ?

$5 \leq 2(2) + 3 = 7$ ✓
$5 > -(2) + 1 = -1$ ✓

Yes, $(2, 5)$ is in the feasible region.

Word Problems with Constraints

The SAT often presents real-world constraints as a system of inequalities.

Example: A farmer has at most 100 acres and wants to plant corn ( $x$ acres) and wheat ( $y$ acres). Corn needs 2 workers per acre and wheat needs 3 workers per acre, and only 240 workers are available.

$x + y \leq 100$ $2x + 3y \leq 240$ $x \geq 0, \quad y \geq 0$

The feasible region shows all valid planting combinations.

Vertices of the Feasible Region

The corners (vertices) of the feasible region are found by solving pairs of boundary equations simultaneously. These are often the points that maximise or minimise an objective.

Bounded vs. Unbounded Regions

Bounded: the feasible region is enclosed (finite area)
Unbounded: the region extends infinitely in some direction

Strategy Tips

Tip 1: Graph Both Inequalities, Then Find the Overlap

Don't try to visualise the overlap mentally. Graph each one clearly and look for where shading overlaps.

Tip 2: Test Points for Each Inequality

To verify a point is in the feasible region, check it against EVERY inequality in the system.

Tip 3: Use Desmos on the SAT

The Desmos calculator can graph systems of inequalities. Enter each inequality and it shows the overlapping region.

Tip 4: For SAT Questions, Focus on the Multiple-Choice Answers

If asked which point is in the feasible region, just test each answer choice against all inequalities. No graphing needed.

Tip 5: Look for the Non-Negative Constraints

Real-world problems usually include $x \geq 0$ and $y \geq 0$ , which restricts the feasible region to the first quadrant.

Worked Example: Example 1

Problem

Which point satisfies both $y \leq x + 4$ and $y \geq 2x - 1$ ?

Test $(1, 3)$ :

$3 \leq 1 + 4 = 5$ ✓
$3 \geq 2(1) - 1 = 1$ ✓

Yes, $(1, 3)$ is a solution.

Solution

Worked Example: SAT-Style Constraint Problem

Problem

A company makes widgets ( $x$ ) and gadgets ( $y$ ). Each widget requires 2 hours of labour; each gadget requires 3 hours. There are at most 120 hours available. At least 10 widgets and 5 gadgets must be made. Write the system of inequalities.

$2x + 3y \leq 120$ $x \geq 10$ $y \geq 5$

Solution

Worked Example: Example 3

Problem

A graph shows a shaded region bounded by $y \leq -x + 8$ , $y \leq 2x + 2$ , and $y \geq 0$ . Find the vertices of the feasible region.

Intersection of $y = -x + 8$ and $y = 2x + 2$ : $-x + 8 = 2x + 2$ → $6 = 3x$ → $x = 2$ , $y = 6$ . Vertex: $(2, 6)$ .

Intersection of $y = -x + 8$ and $y = 0$ : $0 = -x + 8$ → $x = 8$ . Vertex: $(8, 0)$ .

Intersection of $y = 2x + 2$ and $y = 0$ : $0 = 2x + 2$ → $x = -1$ . But if $x \geq 0$ is implied, this vertex is $(0, 2)$ from $y = 2(0) + 2$ .

Solution

Worked Example: Example 4

Problem

The system $x + y \leq 10$ , $x \geq 2$ , $y \geq 3$ defines a feasible region. Which of the following is NOT in the feasible region? A) $(3, 5)$ B) $(4, 6)$ C) $(2, 9)$ D) $(6, 5)$

Test each:

A: $3+5=8 \leq 10$ ✓, $3 \geq 2$ ✓, $5 \geq 3$ ✓
B: $4+6=10 \leq 10$ ✓, $4 \geq 2$ ✓, $6 \geq 3$ ✓
C: $2+9=11 > 10$ ✗
D: $6+5=11 > 10$ ✗

Both C and D fail. If only one answer: C or D (check the question).

Solution

Worked Example: Example 5

Problem

Write a system of inequalities for: a diet requires at least 20g of protein and at most 50g of fat. Food A has 2g protein and 5g fat per serving. Food B has 4g protein and 3g fat per serving.

Let $a$ = servings of A, $b$ = servings of B.

$2a + 4b \geq 20 \quad \text{(protein)}$ $5a + 3b \leq 50 \quad \text{(fat)}$ $a \geq 0, \quad b \geq 0$

Solution

Practice Problems

Problem 1

Which of these points satisfies $y > x - 1$ AND $y < -2x + 8$ : $(1, 2)$ , $(3, 3)$ , $(0, 5)$ ?

Problem 2

A student has at most 6 hours to study. Math takes 1 hour per chapter and English takes 1.5 hours per chapter. Write the constraint inequality.

Problem 3

Graph the system: $y \leq x + 3$ , $y \geq -x + 1$ , $x \geq 0$ . Describe the feasible region.

Problem 4

Find the vertices of the region defined by $x + y \leq 6$ , $x \geq 0$ , $y \geq 0$ .

Problem 5

A budget allows spending at most $200 on shirts ($25 each) and pants ($40 each), buying at least 3 items total. Write the system.

Problem 6

Is $(4, 3)$ a solution to: $2x + y \leq 12$ , $x - y \geq 0$ , $y \geq 1$ ?

Want to check your answers and get step-by-step solutions?

Common Mistakes

Checking only one inequality. A point must satisfy ALL inequalities in the system.
Mixing up shading direction. Graph each inequality carefully before looking for overlap.
Forgetting non-negative constraints. In word problems, $x \geq 0$ and $y \geq 0$ are usually implied.
Not finding all vertices. Solve all pairs of boundary equations that border the feasible region.
Confusing boundary inclusion. Points on a dashed boundary are NOT in the feasible region.

Frequently Asked Questions

How many inequalities can a system have?

As many as needed. SAT problems typically have 2–4 inequalities.

Can the feasible region be empty?

Yes, if the constraints are contradictory (e.g., $y > 5$ and $y < 3$ ).

Do I need to graph on the SAT?

Not always. Often you can test points directly or use Desmos.

What's the connection to word problems?

Many SAT word problems describe constraints (budget, time, resources) that form a system of inequalities.

Can I solve systems of inequalities algebraically?

Not in the same way as systems of equations. The solution is a region, not a point. You can test specific points algebraically.

Key Takeaways

✓
System solution = overlapping region where all inequalities are satisfied.
✓
Test points against all inequalities to verify membership.
✓
Word problems translate constraints into inequalities.
✓
Vertices are found by solving pairs of boundary equations.
✓
Use Desmos on the SAT to visualise systems.
✓
Every inequality must be satisfied — check each one.

Tags:

#sat#math#algebra#inequalities#systems

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Core Concepts

Graphing a System of Inequalities

Identifying Points in the Feasible Region

Word Problems with Constraints

Vertices of the Feasible Region

Bounded vs. Unbounded Regions

Strategy Tips

Tip 1: Graph Both Inequalities, Then Find the Overlap

Tip 2: Test Points for Each Inequality

Tip 3: Use Desmos on the SAT

Tip 4: For SAT Questions, Focus on the Multiple-Choice Answers

Tip 5: Look for the Non-Negative Constraints

Worked Example: Example 1

Worked Example: SAT-Style Constraint Problem

Worked Example: Example 3

Worked Example: Example 4

Worked Example: Example 5

Practice Problems

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Common Mistakes

Frequently Asked Questions

Key Takeaways

Related Topics

Parallel Structure

Verb Tense and Consistency

Modifier Placement and Dangling Modifiers

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