# 5.6 Graphing systems of linear inequalities

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By the end of this section, you will be able to:
• Determine whether an ordered pair is a solution of a system of linear inequalities
• Solve a system of linear inequalities by graphing
• Solve applications of systems of inequalities

Before you get started, take this readiness quiz.

1. Graph $x>2$ on a number line.
If you missed this problem, review [link] .
2. Solve the inequality $2a<5a+12$ .
If you missed this problem, review [link] .
3. Determine whether the ordered pair $\left(3,\frac{1}{2}\right)$ is a solution to the system $\left\{\begin{array}{c}x+2y=4\hfill \\ y=6x\hfill \end{array}$ .
If you missed this problem, review [link]

## Determine whether an ordered pair is a solution of a system of linear inequalities

The definition of a system of linear inequalities is very similar to the definition of a system of linear equations.

## System of linear inequalities

Two or more linear inequalities grouped together form a system of linear inequalities    .

A system of linear inequalities looks like a system of linear equations, but it has inequalities instead of equations. A system of two linear inequalities is shown below.

$\left\{\begin{array}{c}x+4y\ge 10\hfill \\ 3x-2y<12\hfill \end{array}$

To solve a system of linear inequalities, we will find values of the variables that are solutions to both inequalities. We solve the system by using the graphs of each inequality and show the solution as a graph. We will find the region on the plane that contains all ordered pairs $\left(x,y\right)$ that make both inequalities true.

## Solutions of a system of linear inequalities

Solutions of a system of linear inequalities are the values of the variables that make all the inequalities true.

The solution of a system of linear inequalities is shown as a shaded region in the x-y coordinate system that includes all the points whose ordered pairs make the inequalities true.

To determine if an ordered pair is a solution to a system of two inequalities, we substitute the values of the variables into each inequality. If the ordered pair makes both inequalities true, it is a solution to the system.

Determine whether the ordered pair is a solution to the system. $\left\{\begin{array}{c}x+4y\ge 10\hfill \\ 3x-2y<12\hfill \end{array}$

(−2, 4) (3,1)

## Solution

1. Is the ordered pair (−2, 4) a solution?

The ordered pair (−2, 4) made both inequalities true. Therefore (−2, 4) is a solution to this system.

1. Is the ordered pair (3,1) a solution?

The ordered pair (3,1) made one inequality true, but the other one false. Therefore (3,1) is not a solution to this system.

Determine whether the ordered pair is a solution to the system.
$\left\{\begin{array}{c}x-5y>10\hfill \\ 2x+3y>-2\hfill \end{array}$

$\left(3,-1\right)$ $\left(6,-3\right)$

no yes

Determine whether the ordered pair is a solution to the system.
$\left\{\begin{array}{c}y>4x-2\hfill \\ 4x-y<20\hfill \end{array}$

$\left(2,1\right)$ $\left(4,-1\right)$

no no

## Solve a system of linear inequalities by graphing

The solution to a single linear inequality is the region on one side of the boundary line that contains all the points that make the inequality true. The solution to a system of two linear inequalities is a region that contains the solutions to both inequalities. To find this region, we will graph each inequality separately and then locate the region where they are both true. The solution is always shown as a graph.

## How to solve a system of linear inequalities

Solve the system by graphing.

$\left\{\begin{array}{c}y\ge 2x-1\hfill \\ y

## Solution

What is the lcm of 340
How many numbers each equal to y must be taken to make 15xy
15x
Martin
15x
Asamoah
15x
Hugo
1y
Tom
1y x 15y
Tom
find the equation whose roots are 1 and 2
(x - 2)(x -1)=0 so equation is x^2-x+2=0
Ranu
I believe it's x^2-3x+2
NerdNamedGerg
because the X's multiply by the -2 and the -1 and than combine like terms
NerdNamedGerg
find the equation whose roots are -1 and 4
Ans = ×^2-3×+2
Gee
find the equation whose roots are -2 and -1
(×+1)(×-4) = x^2-3×-4
Gee
there's a chatting option in the app wow
Nana
That's cool cool
Nana
Nice to meet you all
Nana
you too.
Joan
😃
Nana
Hey you all there are several Free Apps that can really help you to better solve type Equations.
Debra
Debra, which apps specifically. ..?
Nana
am having a course in elementary algebra ,any recommendations ?
samuel
Samuel Addai, me too at ucc elementary algebra as part of my core subjects in science
Nana
me too as part of my core subjects in R M E
Ken
at ABETIFI COLLEGE OF EDUCATION
Ken
ok great. Good to know.
Joan
5x + 1/3= 2x + 1/2
sanam
Plz solve this
sanam
5x - 3x = 1/2 - 1/3 2x = 1/6 x = 1/12
Ranu
Thks ranu
sanam
a trader gains 20 rupees loses 42 rupees and then gains ten rupees Express algebraically the result of his transactions
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
Kim is making eight gallons of punch from fruit juice and soda. The fruit juice costs $6.04 per gallon and the soda costs$4.28 per gallon. How much fruit juice and how much soda should she use so that the punch costs $5.71 per gallon? Mohamed Reply (a+b)(p+q+r)(b+c)(p+q+r)(c+a) (p+q+r) muhammad Reply 4x-7y=8 2x-7y=1 what is the answer? Ramil Reply x=7/2 & y=6/7 Pbp x=7/2 & y=6/7 use Elimination Debra true bismark factoriz e usman 4x-7y=8 X=7/4y+2 and 2x-7y=1 x=7/2y+1/2 Peggie Ok cool answer peggie Frank thanks Ramil copy and complete the table. x. 5. 8. 12. then 9x-5. to the 2nd power+4. then 2xto the second power +3x Sandra Reply What is c+4=8 Penny Reply 2 Letha 4 Lolita 4 Rich 4 thinking C+4=8 -4 -4 C =4 thinking I need to study Letha 4+4=8 William During two years in college, a student earned$9,500. The second year, she earned $500 more than twice the amount she earned the first year. Nicole Reply 9500=500+2x Debra 9500-500=9000 9000÷2×=4500 X=4500 Debra X + Y = 9500....... & Y = 500 + 2X so.... X + 500 + 2X = 9500, them X = 3000 & Y = 6500 Pbp Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles. Josh Reply Reiko needs to mail her Christmas cards and packages and wants to keep her mailing costs to no more than$500. The number of cards is at least 4 more than twice the number of packages. The cost of mailing a card (with pictures enclosed) is $3 and for a package the cost is$7.
hey
Juan
Sup
patrick
The sum of two numbers is 155. The difference is 23. Find the numbers
The sum of two numbers is 155. Their difference is 23. Find the numbers
Michelle
The difference between 89 and 66 is 23
Ciid
Joy is preparing 20 liters of a 25% saline solution. She only has 40% and 10% solution in her lab. How many liters of the 40% and how many liters of the 10% should she mix to make the 25% solution?
hello
bismark
I need a math tutor BAD
Stacie
Me too
Letha
me too
Xavier
ok
Bishal
teet
Bishal
my answer is y10 that right
misty