Systems of inequalities
Systems of inequalities
Combine multiple constraints, identify the overlap, and reason about feasible regions on the coordinate plane.
The big idea
A system of inequalities asks for points that satisfy every condition at the same time.
1. Each inequality creates a region
A single linear inequality divides the coordinate plane into two half-planes. A system combines those regions.
The boundary is solid, and the solution is above the line.
The boundary is solid, and the solution is below the line.
2. Graph the boundaries first
Replace each inequality with equality to obtain its boundary line. Then decide whether the boundary is solid or dashed.
| Inequality | Boundary equation | Boundary |
|---|---|---|
| \(y\ge x-1\) | \(y=x-1\) | Solid |
| \(y<-x+5\) | \(y=-x+5\) | Dashed |
3. Shade each condition separately
Determine the solution side for each inequality before looking for the overlap. If the inequality is solved for \(y\), greater-than means above and less-than means below.
4. The overlap is the solution
A point belongs to the system only where both shaded conditions are true. The common region is the solution set.
The green region is the set of points satisfying both inequalities.
5. Find a boundary intersection
When the two boundary lines intersect, solve the boundary equations together.
The intersection is \((3,2)\). Whether that point belongs to the system depends on whether each original inequality includes its boundary.
6. Test a point against the whole system
A point must pass every inequality.
7. Solid and dashed boundaries can occur together
Each inequality controls its own boundary. A system can therefore contain one solid line and one dashed line.
The first boundary is included; the second is excluded.
8. Contexts create systems of constraints
SAT questions may describe several limits that must hold simultaneously.
Each inequality represents one restriction. The feasible region is the overlap of all four conditions.
9. Maximum and minimum questions
If a system creates a bounded feasible region, a linear quantity may reach its greatest or least value at a boundary point or corner.
10. Parallel boundaries can eliminate the solution set
Parallel boundaries never intersect. If their inequality directions require opposite sides with no common region, the system has no solution.
A point cannot be at or below \(2x+1\) and at or above \(2x+5\) at the same time.
11. Parameter questions
If the SAT gives a specific point and asks about a parameter, substitute the point into every inequality first.
At \((0,0)\), these become \(0
12. A reliable SAT workflow
Visual practice
Each visual question is self-contained: the equations needed to interpret the graph are shown directly on the graph.
The graph shows the system \(\displaystyle y\ge x-1\) and \(\displaystyle y\le -x+5\). Which point is in the shaded overlap?
- A) \((2,2)\)
- B) \((0,-2)\)
- C) \((4,4)\)
- D) \((5,0)\)
The graph shows the system \(\displaystyle y>x+2\) and \(\displaystyle y
The graph shows the system \(\displaystyle y\le x+4\) and \(\displaystyle y\ge -x+2\). Is \(P=(2,3)\) a solution?
- A) Yes, because it satisfies both inequalities.
- B) No, because it is above both boundaries.
- C) No, because it is below both boundaries.
- D) Yes, because every point on either boundary is a solution.
The feasible region is defined by \(\displaystyle x\ge1\), \(\displaystyle y\ge2\), and \(\displaystyle x+y\le6\). What is the greatest possible value of \(x\)?
- A) 1
- B) 2
- C) 4
- D) 6
The graph shows the system \(\displaystyle y\ge x-2\) and \(\displaystyle y<-x+4\). Is the intersection point of the boundary lines part of the solution set?
- A) Yes, because both boundaries intersect there.
- B) No, because one inequality is strict.
- C) Yes, because the solid boundary is included.
- D) No, because neither boundary is included.
Additional practice
Original SATMath800 practice. These questions vary the representation: algebraic systems, points, boundaries, contexts, and feasible-region reasoning.
Which point satisfies both \(\displaystyle y\ge x-2\) and \(\displaystyle y\le-x+6\)?
- A) \((0,-3)\)
- B) \((1,0)\)
- C) \((2,5)\)
- D) \((4,1)\)
The system \(\displaystyle y\le2x+1\) and \(\displaystyle y\ge2x+5\) has which type of solution set?
- A) Exactly one point
- B) A finite segment
- C) Infinitely many points
- D) No points
For the system \(\displaystyle y\ge x+2\) and \(\displaystyle y\le-x+8\), the boundary lines intersect at which point?
- A) \((2,4)\)
- B) \((3,5)\)
- C) \((4,6)\)
- D) \((5,3)\)
Which system represents points that are above or on \(y=2x-1\) and below \(y=-x+8\)?
- A) \(y\le2x-1\) and \(y\ge-x+8\)
- B) \(y\ge2x-1\) and \(y\le-x+8\)
- C) \(y>2x-1\) and \(y<-x+8\)
- D) \(y\le2x-1\) and \(y\le-x+8\)
In the system \(\displaystyle y<-x+a\) and \(\displaystyle y>x+b\), the point \((0,0)\) is a solution. Which relationship must be true?
- A) \(a>b\)
- B) \(b>a\)
- C) \(a=0\)
- D) \(b=0\)
A school can use at most 26 total hours for two activities. If \(x\) is science-lab time and \(y\) is art-studio time, which system also requires at least 8 hours of science-lab time?
- A) \(x+y\ge26,\ x\ge8\)
- B) \(x+y\le26,\ x\ge8\)
- C) \(x+y\le26,\ x\le8\)
- D) \(x+y\ge26,\ x\le8\)
The system \(\displaystyle y\ge-2x+4\) and \(\displaystyle y\le x+1\) has a feasible region. Which point is on both boundary lines?
- A) \((0,1)\)
- B) \((1,2)\)
- C) \((2,3)\)
- D) \((3,4)\)
A company makes two products. Product A requires 2 machine-hours and product B requires 3 machine-hours. The factory has at most 60 machine-hours. Which inequality is one of the constraints?
- A) \(2x+3y\ge60\)
- B) \(2x+3y\le60\)
- C) \(2x+3y<0\)
- D) \(2x+3y=60\)
The boundary lines of a feasible system intersect at \((150,750)\). If every feasible point satisfies \(y\le-15x+3000\) and \(y\le5x\), what is the greatest possible value of \(y\)?
- A) 150
- B) 600
- C) 750
- D) 3000
Which point satisfies both \(\displaystyle 2x+y\le10\) and \(\displaystyle x+2y\le8\)?
- A) \((2,2)\)
- B) \((4,3)\)
- C) \((5,2)\)
- D) \((1,5)\)
Which system has a solution set that includes the boundary \(y=x+3\) but excludes the boundary \(y=-x+7\)?
- A) \(y>x+3,\ y\le-x+7\)
- B) \(y\ge x+3,\ y<-x+7\)
- C) \(y\le x+3,\ y\ge-x+7\)
- D) \(y>x+3,\ y>-x+7\)
For \(\displaystyle y\ge2x-4\) and \(\displaystyle y\le-2x+8\), what is the \(y\)-coordinate of the intersection of the boundaries?
- A) 2
- B) 4
- C) 6
- D) 8
A point is a solution of a system of three inequalities. What must be true?
- A) It satisfies at least one inequality.
- B) It satisfies exactly two inequalities.
- C) It satisfies all three inequalities.
- D) It lies on all three boundary lines.
A feasible region is bounded by \(x=1\), \(y=2\), and \(x+y=6\), with \(x\ge1\), \(y\ge2\), and \(x+y\le6\). What is the greatest possible value of \(x\)?
- A) 1
- B) 2
- C) 4
- D) 6
Which system represents \(x\) and \(y\) both nonnegative and their sum at least 12?
- A) \(x\le0,\ y\le0,\ x+y\le12\)
- B) \(x\ge0,\ y\ge0,\ x+y\ge12\)
- C) \(x\ge0,\ y\le0,\ x+y\ge12\)
- D) \(x\le0,\ y\ge0,\ x+y\le12\)
For the system \(y\le3x+2\) and \(y\ge3x-4\), which statement is true?
- A) The system has no solution.
- B) The boundaries intersect at one point.
- C) The system has infinitely many solutions.
- D) The solution is only the origin.
For the system \(y\ge x+1\) and \(y\le4\), which \(x\)-values can occur in a solution?
- A) \(x\le3\)
- B) \(x\ge3\)
- C) \(x<4\)
- D) \(x>4\)
A point \((a,b)\) lies on both boundaries \(y=2x+1\) and \(y=-x+7\). What is \(a+b\)?
- A) 6
- B) 7
- C) 8
- D) 9
Which point lies on the boundary \(x+y=10\) and satisfies \(x\ge4\) and \(y\ge3\)?
- A) \((3,7)\)
- B) \((4,6)\)
- C) \((6,5)\)
- D) \((7,4)\)
The feasible region of a system is the overlap of two half-planes. Which operation describes that overlap?
- A) Union
- B) Intersection
- C) Reflection
- D) Translation