Linear inequalities in two variables
Linear inequalities in two variables
Use boundary lines, test points, intercepts, and shaded regions to represent and interpret solutions in the xy-plane.
The big idea
A linear inequality in two variables describes a region of the xy-plane, not usually a single point.
The equation gives the line that separates the two sides.
The inequality selects one side of the boundary line.
1. Start with the boundary line
Replace the inequality symbol with an equals sign. The resulting equation is the boundary line.
This line divides the plane into two regions. The original inequality tells you which region contains the solutions.
2. Solid or dashed?
Points on the boundary are included.
Points on the boundary are excluded.
3. Shade above or below
Once the inequality is written as an inequality in \(y\), the direction of the inequality tells you which side to shade.
Larger \(y\)-values lie above the line.
Smaller \(y\)-values lie below the line.
The dashed boundary is excluded; the region above it is shaded.
The solid boundary is included; the region below it is shaded.
4. When the inequality is not solved for \(y\)
Rewrite the inequality so that \(y\) is isolated. If you divide by a negative number, reverse the inequality symbol.
5. Use a test point when the shading is not obvious
Pick a point that is not on the boundary, substitute its coordinates into the inequality, and use the result to identify the correct side.
Therefore the region containing \((0,0)\) is the solution region.
6. Ordered pairs can be tested directly
A point \((a,b)\) is a solution exactly when its coordinates make the original inequality true.
Use the original inequality symbol in place of the square. Do not test only the boundary equation when the question asks whether a point is a solution.
7. Read intercepts from the boundary
An x-intercept is where the boundary crosses the x-axis, so its y-coordinate is 0. A y-intercept is where the boundary crosses the y-axis, so its x-coordinate is 0.
The y-intercept is where the line crosses the y-axis. The x-intercept is where the line crosses the x-axis.
8. Read a graph backwards
SAT questions may give you the graph first and ask you to determine the inequality. Use four pieces of evidence:
9. Context can create a region
If two variables represent quantities, a linear inequality can describe a constraint on all possible pairs.
Here every nonnegative ordered pair \((x,y)\) satisfying the inequality is a possible combination under the stated constraint. The graph therefore represents many possible combinations rather than one answer.
10. Integer and nonnegative restrictions matter
The algebraic region may contain infinitely many real-number points, but a context can restrict which points are meaningful.
Useful for counts, lengths, amounts, and other quantities that cannot be negative.
Useful when a variable counts people, objects, days, or other discrete items.
11. A reliable SAT strategy
Visual practice
These questions require you to read the coordinate plane. Every graph provides the equation or the axis information needed for the task.
The graph shows the boundary line \(y=x-2\) as a solid line, with the region above the line shaded. Which inequality represents the shaded region?
- A) \(y>x-2\)
- B) \(y\ge x-2\)
- C) \(y<x-2\)
- D) \(y\le x-2\)
The graph shows the dashed boundary \(y=-\frac12x+3\). What is the x-intercept of the boundary line?
- A) \((0,3)\)
- B) \((3,0)\)
- C) \((6,0)\)
- D) \((0,6)\)
The graph shows the boundary \(y=-x+4\) and point \(P=(2,1)\). Is \(P\) a solution of \(y<-x+4\)?
- A) Yes
- B) No
- C) Only if the boundary is solid
- D) Only when \(x<0\)
The graph shows the two boundary lines \(y=x+1\) and \(y=-x+7\). At what point do the boundary lines intersect?
- A) \((2,3)\)
- B) \((3,4)\)
- C) \((4,3)\)
- D) \((3,5)\)
Additional practice
Original SATMath800 practice. The set mixes algebraic, graphical, contextual, and ordered-pair reasoning rather than repeating one question pattern.
Which point is a solution to \(y>2x-3\)?
- A) \((0,-4)\)
- B) \((1,0)\)
- C) \((2,0)\)
- D) \((-1,-6)\)
Which inequality represents the region above \(y=-3x+5\), including the boundary?
- A) \(y<-3x+5\)
- B) \(y\le-3x+5\)
- C) \(y>-3x+5\)
- D) \(y\ge-3x+5\)
Which statement about \(y<\frac12x+4\) is true?
- A) The boundary is solid and the region is above it.
- B) The boundary is dashed and the region is above it.
- C) The boundary is dashed and the region is below it.
- D) The boundary is solid and the region is below it.
Which inequality is equivalent to \(3x+2y\le12\)?
- A) \(y\le-\frac32x+6\)
- B) \(y\ge-\frac32x+6\)
- C) \(y\le\frac32x+6\)
- D) \(y\ge\frac32x-6\)
A point \((a,b)\) lies on the boundary of \(2x-y>7\). Which equation must be true?
- A) \(2a-b>7\)
- B) \(2a-b=7\)
- C) \(2a-b<7\)
- D) \(a-2b=7\)
The inequality \(x-2y\ge6\) is graphed in the xy-plane. Which point is in its solution region?
- A) \((8,1)\)
- B) \((4,0)\)
- C) \((1,-2)\)
- D) \((-1,-2)\)
A line has equation \(4x+y=8\). Which inequality represents the half-plane below this line?
- A) \(4x+y<8\)
- B) \(4x+y>8\)
- C) \(4x+y\le-8\)
- D) \(4x+y\ge8\)
Which point lies on the boundary line of \(5x+2y\le20\)?
- A) \((0,8)\)
- B) \((2,5)\)
- C) \((4,0)\)
- D) \((6,-5)\)
A study plan allows at most 420 minutes. If \(x\) is the number of minutes spent on algebra and \(y\) is the number of minutes spent on geometry, which inequality represents the time constraint?
- A) \(x+y\le420\)
- B) \(xy\le420\)
- C) \(x+y\ge420\)
- D) \(xy\ge420\)
For \(y\ge-2x+1\), which description is correct?
- A) Dashed boundary; shade below.
- B) Dashed boundary; shade above.
- C) Solid boundary; shade below.
- D) Solid boundary; shade above.
Which point satisfies both \(y\le x+2\) and \(y>-x+4\)?
- A) \((0,1)\)
- B) \((1,3)\)
- C) \((2,3)\)
- D) \((4,0)\)
A constraint is written as \(2x+5y\ge30\). Which statement about its boundary is correct?
- A) The boundary is \(2x+5y>30\).
- B) The boundary is \(2x+5y=30\).
- C) The boundary is \(2x+5y<30\).
- D) There is no boundary line.
The graph of a linear inequality has a dashed boundary line. Which inequality symbol could describe it?
- A) \(\le\)
- B) \(\ge\)
- C) \(<\)
- D) Either \(\le\) or \(\ge\)
Which inequality describes the half-plane containing \((0,0)\) for the line \(y=3x-5\), with the boundary included?
- A) \(y\ge3x-5\)
- B) \(y\le3x-5\)
- C) \(y>3x-5\)
- D) \(y<3x-5\)
A water tank must hold at least 600 liters. If \(x\) and \(y\) are nonnegative quantities and the total is \(x+y\), which inequality models the requirement?
- A) \(x+y<600\)
- B) \(x+y\le600\)
- C) \(x+y>600\)
- D) \(x+y\ge600\)
Which point satisfies \(6x-3y<0\)?
- A) \((1,1)\)
- B) \((0,1)\)
- C) \((2,4)\)
- D) \((3,1)\)