Solving Systems of Linear Inequalities

Solving Systems of Linear Inequalities

The graph of a system of linear inequalities should show the region of the graph where the points are solutions to all the inequalities in the system.

Here is the system of the graph

Multiple Choice

What is the solution to the graph of the system provided. (Click on the image to view graph)

• (1, -3)

• (6, -4)

• (-2, 3)

• (-6, -4)

When plotting system of inequalities on the graph, you must ask yourself a couple of questions.

-Is the inequality a solid or dashed line? (You are asking this for each equation)

-Is the shade over or shade under?

Example: Plot the system of inequalities on the graph

Let's first take a look at equation 1

 $y>-3x+4$  

Multiple Choice

The inequality symbol

$$\le\ or\ \ge$$  means a dashed line

• True

• False

Multiple Choice

The inequality symbols

$$>\ or\ \ge$$  means shade over. 

• True

• False

Now let's take a look at equation 2.

Multiple Choice

The inequality symbol

$$<\ or\ >$$  means a dashed line.

• True

• False

Multiple Choice

The inequality symbol

$$<\ or\ \le$$  means to shade under.

• True

• False

The final step

Once you have graphed both equations, you will need to identify where they overlap.

Find a point in the solution set.

(4, 0) appears to be in the double shaded area. Verify (4,0) by plugging the values into each equation and see if the result is true.

Multiple Choice

Another solution to the graph of the example is? (Click on the graph to enlarge image)

• 2,1

• -4,2

• 4,2

• -2,1

Multiple Choice

In a system of inequality, the shaded area of where the two graphs overlap mean the points are solutions of all the inequalities in the system.

• True

• False

Multiple Choice

What is the graph to the system of inequalities?

$$y>4x-3$$  
$$y\ge-2x+3$$  

Multiple Choice

Which system of inequalities represents the graph. (Click on image to view graph)

• 
  

  $$4x+3y>9$$

  $$x-3y<6$$
  

• $$4x-3y<9$$
  $$x+3y>6$$
  

• $$4x-3y<9$$
  $$x+3y<6$$
  

• $$4x+3y<9$$
  $$x-3y>6$$
  

How to set up inequalities

Example 1: You can work at most 20 hours next week. You need to earn at least $92 to cover your weekly expenses. Your dog-walking job pays $7.50 per hour and your job as a car wash attendant pays $6 per hour. Write a system of linear inequalities to model the situation.

Breaking up the problem

Example 1: You can work at most (can't go over 20 use less than or equal to symbol) 20 hours next week. You need to earn at least (you can make more than 92 dollars. Use greater than or equal to symbol) $92 to cover your weekly expenses. Your dog-walking job (represented by x) pays $7.50 per hour and your job as a car wash attendant (represented by y) pays $6 per hour. Write a system of linear inequalities to model the situation.

Writing the system with what we know

Multiple Choice

Marsha is buying plants and soil for her garden. The soil cost $4 per bag, and the plants cost $10 each. She wants to buy at least 5 plants and can spend no more than $100. Write a system of linear inequalities to model the situation.

• 
  

  $$4x+10y\ge100$$
  

  $$x\le5$$
  
  

• $$4x-10y\le100$$
  $$x\ge5$$
  

• $$4x+10y\le100$$
  $$y\ge5$$
  

• $$4x-10y\ge100$$
  $$y\le5$$
  

Multiple Choice

Jonah is going to the store to buy candles. Small candles cost $3.50 and large candles cost $5.00. He needs to buy at least 20 candles, and he cannot spend more than $80. Write a system of linear inequalities that represent the situation.

• 
  

  $$x+y\le20$$
  

  $$3.5x+5y\ge80$$
  
  

• $$xy\ge20$$
  $$3.5+5y\le80$$
  

• $$x-y\ge20$$
  $$3.5x+5y\le80$$
  

• $$x+y\ge20$$
  $$3.5x+5y\le80$$
  

Solving Systems by Substitution

look for or create a y= or x=

Multiple Choice

Which correctly sets up the first step to solve the system by substitution?
 
5x + 4y= −14
y = −7x − 15  

• $$5x+4\left(-7x-15\right)=-14$$  

• $$5\left(-7x-15\right)+4y=-14$$  
  

• $$5x+4y=-7x-15$$  
  

• $$5x+7x-15=-14$$  
  

Solving Systems by Elimination

look for or create a zero pair with a variable

Multiple Choice

Which is the correct result after performing the first step of solving by elimination?

4x+9y=28

-4x-y=-28

• $$10y=56$$
  

• $$No\ Solution$$
  

• $$8x+8y=0$$
  

• $$8y=0$$

Multiple Choice

Which point is a solution of the system above?

• (0, 0)

• (0, 1)

• (1, 0)

• (1, 1)

Multiple Choice

Which system of inequalities best describes the graph?

• $$y<2x-3$$ $$y>2x+1$$

• $$y<2x-3$$ $$y<2x+1$$

• $$y>2x-3$$ $$y<2x+1$$

• $$y>2x-3$$ $$y>2x+1$$

Multiple Choice

Which point is not a solution of the system of inequalities?

• (4, -6)

• (3, -7)

• (2, -3)

• (1, -5)

Multiple Choice

Which ordered pair is a solution of the system of linear inequalities?

• (0, 5)

• (-2, -3)

• (-2, 1)

• (-4, 1)

Multiple Choice

Which system of inequalities best describes the graph?

• $$y\ge\frac{1}{3}x-4$$ $$y<-\frac{1}{2}x+3$$

• $$y>\frac{1}{3}x-4$$ $$y\le-\frac{1}{2}x+3$$

• $$y\le\frac{1}{3}x-4$$ $$y>-\frac{1}{2}x+3$$

• $$y<\frac{1}{3}x-4$$ $$y\ge-\frac{1}{2}x+3$$

Multiple Choice

Which is the graph of the system of inequalities?

Multiple Select

Which points are solutions to the system of inequalities? Select all that apply.

• (3, 3)

• (4, 2)

• (0, 0)

• (6, 1)

• (5, 5)

Multiple Choice

Which graph shows the system of linear inequalities?