Back to homeMathematics · Algebra 1 · Change subject

Start for free

Algebra 1 · Mathematics

Equations and Systems · Equations, Inequalities and Systems

Linear Inequalities in Two Variables

Before you start, make sure you know: Systems of equations (Math 8 (Pre-Algebra))

You have $\$20$ at the concession stand at a football game. A hot dog costs $\$4$ and a soda costs $\$2$. You could buy 3 hot dogs and 4 sodas. Or 1 hot dog and 1 soda. Or 2 hot dogs and 5 sodas. There isn't just one answer, there are lots of them. How can you show all the possible orders in one picture?

The answer is an inequality with two variables. Its solution is not a point or a line but a whole region of the plane. (Some problems call the variables "unknowns" and the coordinate plane the "Cartesian plane". It's the same thing.)

Graphing a Linear Inequality

An equation like $2x + y = 4$ has a line as its graph. Change the $=$ to $<$ and you get $2x + y < 4$. Now you want every point where $2x + y$ is less than $4$.

That line splits the plane into two sides, called half-planes. On one side, $2x + y$ is always less than $4$. On the other side, it is always greater than $4$. The line itself is where it equals $4$.

There is a second method. Solve for $y$, and the inequality sign tells you which side to shade.

Form	Line	Shade
$y > mx + b$	dashed	above
$y \ge mx + b$	solid	above
$y < mx + b$	dashed	below
$y \le mx + b$	solid	below

Practice★☆☆

Check which of the following points are a solution to the inequality $3x - 2y \le 6$: $A(0, 0)$ $B(4, 1)$ $C(2, -1)$ $D(-2, -5)$

Check which of the following points are a solution to the inequality $3x - 2y \le 6$:

$A(0, 0)$
$B(4, 1)$
$C(2, -1)$
$D(-2, -5)$

Practice★☆☆

Given the inequality $4x - 5y < 20$, indicate with reasoning whether the points located exactly on the line $4x - 5y = 20$ are part of the solution region.

Given the inequality $4x - 5y < 20$, indicate with reasoning whether the points located exactly on the line $4x - 5y = 20$ are part of the solution region.

Practice★☆☆

Graph the solution region of the inequality $x + 2y > 4$ on the Cartesian plane.

Graph the solution region of the inequality $x + 2y > 4$ on the Cartesian plane.

From a Graph to an Inequality

Now go the other way. You see a shaded region and you need to write its inequality. Three clues give you everything:

The slope and y-intercept of the boundary line give its equation.
A solid or dashed line gives $\le, \ge$ or $<, >$.
The shaded side tells you above ($y > \dots$) or below ($y < \dots$).

Practice★☆☆

Write the linear inequality with two unknowns whose solution region is represented in the following graph. Take into account whether the boundary line is solid or dashed.

Write the linear inequality with two unknowns whose solution region is represented in the following graph. Take into account whether the boundary line is solid or dashed.

Practice★★☆

Find the system of inequalities that defines the shaded region in the following graph:

Find the system of inequalities that defines the shaded region in the following graph:

Systems of Inequalities

Often one condition isn't enough. A system of inequalities has two or more inequalities, and a point must satisfy all of them.

Horizontal and vertical boundaries are easy to shade:

Inequality	Boundary	Shade
$x \ge a$	vertical line $x = a$	right
$x \le a$	vertical line $x = a$	left
$y \ge b$	horizontal line $y = b$	above
$y \le b$	horizontal line $y = b$	below

For example, the first quadrant is the system $x \ge 0$, $y \ge 0$.

Graph $2x + y \le 6$, $x - y \ge -1$, $y \ge 0$

Graph the three boundary lines: $2x + y = 6$, $x - y = -1$ and $y = 0$ (the x-axis). All three signs include equality, so all three lines are solid.

Test the origin in each inequality. $0 \le 6$ is true, so shade below $2x + y = 6$. $0 \ge -1$ is true, so shade the side of $x - y = -1$ containing the origin. And $y \ge 0$ means above the x-axis.

The region where all three shadings overlap is a triangle. Its vertices are where the boundary lines cross in pairs.

Find the vertex where the line $2x + y = 6$ meets the x-axis. (Hint: on the x-axis, $y = 0$.)

???

Substitute $y = 0$: $2x = 6$, so $x = 3$. The vertex is $(3, 0)$. The other two vertices are $(-1, 0)$ (from $x - y = -1$ and $y = 0$) and $\left(\tfrac{5}{3}, \tfrac{8}{3}\right)$ (where the two slanted lines meet). You'll learn to find that last one in the vertices section below.

Practice★☆☆

Determine whether the points $P(1, 3)$, $Q(-1, 2)$ and $R(4, 0)$ belong to the feasible region defined by the following system of inequalities: $x + y \le 5$…

Determine whether the points $P(1, 3)$, $Q(-1, 2)$ and $R(4, 0)$ belong to the feasible region defined by the following system of inequalities:

$x + y \le 5$

$2x - y > -1$

Practice★☆☆

Graph the system of inequalities that defines the fourth quadrant of the Cartesian plane (where the x-coordinate is positive or zero and the y-coordinate is negative or…

Graph the system of inequalities that defines the fourth quadrant of the Cartesian plane (where the x-coordinate is positive or zero and the y-coordinate is negative or zero).

Practice★★☆

Graph the feasible region determined by the following system of inequalities: $2x + y \le 6$ $x - y \ge -1$ $y \ge 0$

Graph the feasible region determined by the following system of inequalities:

$2x + y \le 6$

$x - y \ge -1$

$y \ge 0$

Writing a System from a Region

To describe a bounded region, write one inequality for each side. Test one point inside the region to choose the direction of each sign.

A double bound like $-2 \le x \le 3$ is shorthand for the two inequalities $x \ge -2$ and $x \le 3$. It describes a vertical strip. Add $-1 \le y \le 2$ and the strip is cut into a rectangle. For a rectangle whose edges are included, every inequality uses $\le$ or $\ge$.

Practice★☆☆

Write the system of inequalities whose solution is the vertical strip between $x = -2$ and $x = 3$, including both boundaries.

Write the system of inequalities whose solution is the vertical strip between $x = -2$ and $x = 3$, including both boundaries.

Practice★★★

Write a system of inequalities whose solution is a perfect square centered at the origin of coordinates, with a side of $4$ units, with its sides parallel to the…

Write a system of inequalities whose solution is a perfect square centered at the origin of coordinates, with a side of $4$ units, with its sides parallel to the Cartesian axes, and where the points on the edges are included.

Practice★★★

Determine the system of inequalities that defines the bounded region shown in the following figure:

Determine the system of inequalities that defines the bounded region shown in the following figure:

Word Problems with Inequalities

Limits like budgets, time, space and weight turn into inequalities. Each limit is one inequality, and the possible choices are the feasible region.

Choosing the right sign is half the work:

Words	Symbol
at least, no less than, minimum of	$\ge$
at most, no more than, does not exceed, maximum of	$\le$
more than, exceeds	$>$
fewer than, less than	$<$

Back to the concession stand

Hot dogs cost $\$4$ each, sodas cost $\$2$ each, and you have $\$20$. You want at least one hot dog. Let $x$ = hot dogs and $y$ = sodas.

The budget says the total cost is at most $\$20$: $4x + 2y \le 20$. "At least one hot dog" is $x \ge 1$. You can't buy a negative number of sodas, so $y \ge 0$.

The system is $4x + 2y \le 20$, $x \ge 1$, $y \ge 0$. The feasible region is a triangle. Because you can only buy whole items, only the points with whole-number coordinates count.

Check $(3, 4)$: $4(3) + 2(4) = 20 \le 20$ ✓, so it works (exactly $\$20$). Check $(4, 3)$: $16 + 6 = 22 > 20$ ✗, so it's too expensive.

Name two other combinations that are in the feasible region.

???

Many answers work. For example, $(1, 1)$: cost $4 + 2 = 6$. Or $(2, 5)$: cost $8 + 10 = 18$. Any whole-number point inside the triangle works. So the first question has a whole picture as its answer, not just one number.

Practice★☆☆

Translate the following statement into a linear inequality with two unknowns: "Triple the weight of a large box ($x$) plus double the weight of a small box ($y$) does…

Translate the following statement into a linear inequality with two unknowns: "Triple the weight of a large box ($x$) plus double the weight of a small box ($y$) does not exceed $50$ lb".

Practice★★☆

A workshop manufactures bicycles and tricycles. Each bicycle requires $2$ wheels and each tricycle $3$ wheels. There are $60$ wheels left in the warehouse. Furthermore,…

A workshop manufactures bicycles and tricycles. Each bicycle requires $2$ wheels and each tricycle $3$ wheels. There are $60$ wheels left in the warehouse. Furthermore, it is known that the number of bicycles manufactured must be at least double that of tricycles. Set up the system of inequalities that models this situation.

Practice★★★

A transport company has type A trucks (capacity of $3$ tons) and type B trucks (capacity of $5$ tons). They need to transport at least $45$ tons of goods. Due to staff…

A transport company has type A trucks (capacity of $3$ tons) and type B trucks (capacity of $5$ tons). They need to transport at least $45$ tons of goods. Due to staff restrictions, they cannot use more than $12$ trucks in total. Set up the system of inequalities and check analytically whether it is possible to carry out the transport using $4$ type A trucks and $7$ type B trucks.

Vertices and Unknown Coefficients

A vertex of the feasible region is where two boundary lines meet. To find it, solve the two boundary equations as a system, by substitution or by elimination. (Review the Math 8 page above if you need it.)

A point on a line satisfies its equation. A point in a solution region satisfies its inequality. Both facts help when a coefficient is unknown.

Practice★★☆

Find the values of the parameter $k$ for which the point $A(-1, 4)$ belongs to the solution region of the inequality $kx - 2y \le 3$.

Find the values of the parameter $k$ for which the point $A(-1, 4)$ belongs to the solution region of the inequality $kx - 2y \le 3$.

Practice★★★

Given the system of inequalities: $x + 2y \le 8$ $ax - y \ge 1$ Calculate the value of the parameter $a$ so that the point $P(3, 2)$ is a vertex of the feasible region…

Given the system of inequalities:

$x + 2y \le 8$

$ax - y \ge 1$

Calculate the value of the parameter $a$ so that the point $P(3, 2)$ is a vertex of the feasible region (that is, it is located exactly at the intersection of the two boundary lines).

Practice★★★

Graph the following system to solve it and find analytically the coordinates of the vertices of the feasible region: $y \le 2x + 2$ $y \le -x + 5$ $y \ge 1$

Graph the following system to solve it and find analytically the coordinates of the vertices of the feasible region:

$y \le 2x + 2$

$y \le -x + 5$

$y \ge 1$

Inequalities with a Parabola

When the boundary is a parabola instead of a line, the method is the same. Graph the curve, then test a point to see which side to shade. Use a solid curve for $\le, \ge$ and a dashed curve for $<, >$.

Practice★★☆

Graph the solution of the quadratic inequality with two unknowns $y \ge x^2 - 4x + 3$.

Graph the solution of the quadratic inequality with two unknowns $y \ge x^2 - 4x + 3$.

Practice★★★

Graph the region of the plane defined by the following mixed system (a quadratic inequality and a linear inequality): $y \le -x^2 + 2x + 3$ $x - y \le 1$

Graph the region of the plane defined by the following mixed system (a quadratic inequality and a linear inequality):

$y \le -x^2 + 2x + 3$

$x - y \le 1$

Summary

Let's practice! 8 medium exercises in this section

★★☆ To organize a party, bottles of soft drink are bought at $\$2…★★☆ Find the system of inequalities that defines the shaded region in the following graph:

Let's practice! 8 hard exercises in this section

★★★ A transport company has type A trucks (capacity of $3$ tons) and type B trucks (capacity…★★★ Determine the system of inequalities that defines the bounded region shown in the…★★★ Graph the region of the plane defined by the following mixed system (a quadratic…★★★ Chickens and pigs are raised on a farm. It is known that the maximum number of animals…

See all the exercises in this section