4.7 Graphs of linear inequalities

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By the end of this section, you will be able to:

Verify solutions to an inequality in two variables
Recognize the relation between the solutions of an inequality and its graph
Graph linear inequalities

Before you get started, take this readiness quiz.

Solve: $4 x + 3 > 23$ .
If you missed this problem, review [link] .
Translate from algebra to English: $x < 5$ .
If you missed this problem, review [link] .
Evaluate $3 x - 2 y$ when $x = 1$ , $y = −2$ .
If you missed this problem, review [link] .

Verify solutions to an inequality in two variables

We have learned how to solve inequalities in one variable. Now, we will look at inequalities in two variables. Inequalities in two variables have many applications. If you ran a business, for example, you would want your revenue to be greater than your costs—so that your business would make a profit.

Linear inequality

A linear inequality is an inequality that can be written in one of the following forms:

\begin{matrix} A x + B y > C & A x + B y \geq C & A x + B y < C & A x + B y \leq C \end{matrix}

where $A and B$ are not both zero.

Do you remember that an inequality with one variable had many solutions? The solution to the inequality $x > 3$ is any number greater than 3. We showed this on the number line by shading in the number line to the right of 3, and putting an open parenthesis at 3. See [link] .

The figure shows a number line extending from negative 5 to 5. A parenthesis is shown at positive 3 and an arrow extends form positive 3 to positive infinity.

Similarly, inequalities in two variables have many solutions. Any ordered pair $(x, y)$ that makes the inequality true when we substitute in the values is a solution of the inequality.

Solution of a linear inequality

An ordered pair $(x, y)$ is a solution of a linear inequality if the inequality is true when we substitute the values of x and y .

Determine whether each ordered pair is a solution to the inequality $y > x + 4$ :

ⓐ $(0, 0)$ ⓑ $(1, 6)$ ⓒ $(2, 6)$ ⓓ $(−5, −15)$ ⓔ $(−8, 12)$

ⓐ

$(0, 0)$

Simplify.
So, $(0, 0)$ is not a solution to $y > x + 4$ .
ⓑ

$(1, 6)$

Simplify.
So, $(1, 6)$ is a solution to $y > x + 4$ .
ⓒ

$(2, 6)$

Simplify.
So, $(2, 6)$ is not a solution to $y > x + 4$ .
ⓓ

$(−5, −15)$

Simplify.
So, $(−5, −15)$ is not a solution to $y > x + 4$ .
ⓔ

$(−8, 12)$

Simplify.
So, $(−8, 12)$ is a solution to $y > x + 4$ .

Got questions? Get instant answers now!

Determine whether each ordered pair is a solution to the inequality $y > x - 3$ :

ⓐ $(0, 0)$ ⓑ $(4, 9)$ ⓒ $(−2, 1)$ ⓓ $(−5, −3)$ ⓔ $(5, 1)$

ⓐ yes ⓑ yes ⓒ yes ⓓ yes ⓔ no

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Determine whether each ordered pair is a solution to the inequality $y < x + 1$ :

ⓐ $(0, 0)$ ⓑ $(8, 6)$ ⓒ $(−2, −1)$ ⓓ $(3, 4)$ ⓔ $(−1, −4)$

ⓐ yes ⓑ yes ⓒ no ⓓ no ⓔ yes

Got questions? Get instant answers now!

Recognize the relation between the solutions of an inequality and its graph

Now, we will look at how the solutions of an inequality relate to its graph.

Let’s think about the number line in [link] again. The point $x = 3$ separated that number line into two parts. On one side of 3 are all the numbers less than 3. On the other side of 3 all the numbers are greater than 3. See [link] .

The solution to $x > 3$ is the shaded part of the number line to the right of $x = 3$ .

Similarly, the line $y = x + 4$ separates the plane into two regions. On one side of the line are points with $y < x + 4$ . On the other side of the line are the points with $y > x + 4$ . We call the line $y = x + 4$ a boundary line.

Boundary line

The line with equation $A x + B y = C$ is the boundary line that separates the region where $A x + B y > C$ from the region where $A x + B y < C$ .

For an inequality in one variable, the endpoint is shown with a parenthesis or a bracket depending on whether or not $a$ is included in the solution:

Questions & Answers

differentiate between demand and supply giving examples

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Lambiv

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appreciation

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In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,

Ezea

What is ceteris paribus?

Shukri Reply

other things being equal

AI-Robot

When MP₁ becomes negative, TP start to decline. Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab

Kelo

Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)

Kelo

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Shukri

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what is monopoly mean?

Habtamu Reply

What is different between quantity demand and demand?

Shukri Reply

Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded

Ezea

Shukri

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Economic growth as an increase in the production and consumption of goods and services within an economy.but Economic development as a broader concept that encompasses not only economic growth but also social & human well being.

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Jabir

What do you think is more important to focus on when considering inequality ?

Abdisa Reply

any question about economics?

Awais Reply

sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏

Asui

it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs

Awais

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Asui

In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p

Cornelius

Suppose a consumer consuming two commodities X and Y has The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50. A,Calculate quantities of x and y which maximize utility. B,Calculate value of Lagrange multiplier. C,Calculate quantities of X and Y consumed with a given price. D,alculate optimum level of output .

Feyisa Reply

Answer

Feyisa

Jabir

the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema

Gsbwnw Reply

suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product

Abdureman

types of unemployment

Yomi Reply

What is the difference between perfect competition and monopolistic competition?

Mohammed

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Source: OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2

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$(0, 0)$

Simplify.	So, $(0, 0)$ is not a solution to $y > x + 4$ .

$(1, 6)$

Simplify.	So, $(1, 6)$ is a solution to $y > x + 4$ .

$(2, 6)$

Simplify.	So, $(2, 6)$ is not a solution to $y > x + 4$ .

$(−5, −15)$

Simplify.	So, $(−5, −15)$ is not a solution to $y > x + 4$ .

$(−8, 12)$

Simplify.	So, $(−8, 12)$ is a solution to $y > x + 4$ .