Understanding of Equations and Inequalities

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Equations

&

Inequalities

Review

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Solving Linear

Equations

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Multiple Choice

Solve the following equation,
x + 7 = 28

• x = 35

• x = 4

• x = 196

• x = 21

Multiple Choice

5(2x - 7) = 80

• x = 11.5

• x = 115

• x = 12

• x = 16

Multiple Choice

Solve:
-4x = 24

• 20

• -20

• 6

• -6

Multiple Choice

Solve:
7x - 3x - 8 = 24

• 8

• -8

• 3.2

• 4

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Systems

of

Equations

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System of Equation

System of Equations are two or more

equations working together.

Example:
{

y = -1/2x + 4
y = x - 2

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Types of Solutions for Systems of Equations

❏ One Solution
❏ No Solution
❏ Infinitely Many Solution

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Types of Solutions for Systems of Equations

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One Solution
Demos

Steps

1. Graph the first linear

equation.

2. Graph the second linear

equation.

3. Find the point of

intersection.

Multiple Choice

If a system of linear equations has one solution, what does this mean about the two lines? 

• Parallel lines

• the same line 

• Intersecting lines

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No Solution
{

y = -6x - 3
y = -6x + 1

Since the graphs

will never intersect,
there is no solution.

Steps

1.

Graph the first linear equation.

2.

Graph the second linear

equation.

3.

Find the point of intersection.

Check

If you see that the lines of the

equations are parallel (i.e. same
slope), then there is no solution.

Multiple Choice

If a system of equations has no solution, what does the graph look like? 

• intersecting lines

• parallel lines

• skew lines

• intersecting lines

Multiple Choice

When you graph the exact same equation twice,

• you will have no solution. 

• you will have one solution.

• you will have infinite solutions. 

• you will graph a giraffe. 

Multiple Choice

• Choice A

• Choice B

• Choice C

• Choice D

Multiple Choice

What is the solution?

• A

• B

• C

• D

Multiple Choice

What is the solution to the system?

• (0, 3)

• (1, -1)

• (-3, 1)

• (1, 3)

Multiple Choice

Solve for x and y.
y = 2/3x - 2
y = -x + 3

• (0,3)

• (0,-3)

• (3,0)

• (-3,0)

Multiple Choice

What is the value of the y-coordinate of the solution to the system of equations 
x-2y=1
x+4y=7 

• 1

• -1

• 3

• 4

Fill in the Blank

Solve the system. Record your answer as (x,y).

(

,

)

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One Variable
Inequalities

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One Variable Inequalities

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Inequality Brothers

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Step 2

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Match

Match the following inequality signs with the correct description.

• $$<$$  

• $$>$$  

• $$\le$$  

• $$\ge$$  

Less than

Greater than

Less than or equal to

Greater than or equal to

• Less than

• Greater than

• Less than or equal to

• Greater than or equal to

Multiple Choice

What is the solution set for the inequality?

$$-6x+9>7x+74$$  

• $$x<5$$  

• $$x<-5$$  

• $$x>5$$  

• $$x>-5$$  

Match

Match the following inequalities to the number line that best represents the solution set.

  

 

 

  

• $$1<x$$   

• $$x>1$$  

• $$x\le1$$  

• $$1\ge x$$   

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Two Variable
Inequalities

Multiple Choice

Which of the following inequalities matches the graph?

• y ≤ 3/4x + 5

• y ≥ 3/4x + 5

• y < 3/4x + 5

• y > 3/4x + 5

Multiple Choice

Which of the following inequalities matches the graph?

• y > 2x - 2

• y < 2x - 2

• y ≥ 2x - 2

• y ≤ 2x - 2

Multiple Choice

Which of the following inequalities matches the graph?

• y ≥ x + 2

• y ≤ x + 2

• y > x + 2

• y < x + 2

Multiple Choice

Which of the following inequalities matches the graph?

• y < -4x - 1

• y > -4x - 1

• y ≤ -4x - 1

• y ≥ -4x - 1

Multiple Choice

Which of the following inequalities matches the graph?

• y ≤ 3/4x + 5

• y ≥ 3/4x + 5

• y < 3/4x + 5

• y > 3/4x + 5

Multiple Choice

Which of the following inequalities matches the graph?

• x > -2

• x < -2

• y > -2

• y < -2

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Systems of
Inequalities

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What is a system of Inequality

A System of Inequality is:
two or more linear inequalities with
the same variables being
compared.

SOLUTIONS to a SYSTEM OF
INEQUALITIES are:

all points within the overlapping
shaded region that makes each
inequality in system true.

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Inequality Brothers

Multiple Choice

Which system of inequalities matches the graph?

• y > 2x - 4
  y < 2

• y < 2x - 4
  y < 2

• y > 2x - 4
  y <  2 

• y < 2x - 4
  y <   -2

Multiple Choice

An inequality using a > or a < has a ___________ line

• invisible

• solid

• dashed

Multiple Choice

Solve the system of inequalities by graphing.

Multiple Select

Pick ALL correct answers.

Which are solutions to the system?

• (-5,4)

• (-5,-2)

• (0,10)

• (5,10)

• (0,-5)

Multiple Choice

The graph shows which system?

• 
  

  $$y>\frac{2}{3}x+1$$
  

  
  $$y\le-5x+10$$
  

• $$y<\frac{2}{3}x+1$$
  $$y\ge-5x+10$$
  

• $$y<\frac{2}{3}x+1$$
  $$y\le-5x+10$$
  

• $$y>\frac{2}{3}x+1$$
  $$y\ge-5x+10$$
  

Multiple Choice

Which system is graphed?

• 
  

  $$y>-\frac{1}{4}x-3$$  

  
  $$y>2x+6$$  
  

• $$y>\frac{1}{4}x-3$$  
  $$y\le-2x+6$$  
  

• $$y\le-2x+6$$  
  $$y<-\frac{1}{4}x-3$$  
  

• $$y\le-2x+6$$  
  $$y>-\frac{1}{4}x-3$$  
  

Multiple Choice

Given the system, determine a solution.

• No solution

• Infinite solutions

• (-5, 5)

• (5, -5)

Multiple Choice

What inequality does the number line graph represent?

• x > 5

• x < −5

• x ≥ 5

• x ≥ −5

Multiple Choice

Select the inequality shown

• x > 5

• x ≥ 5

• x < 5

• x ≤ 5

Multiple Choice

Which graph represents the system of inequalities?

$$y\ge\frac{3}{4}x-2$$  

$$y<-\frac{1}{3}x+2$$  

Multiple Select

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

•  (0,0)

•  (3,1)

•  (-4,-2)

•  (3,-2)

• (-1,3)

Multiple Select

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

•  (0,0)

•  (4,4)

•  (0,2)

•  (-4,0)

• (2,-4)

Multiple Choice

Is (3,-4) a solution to this system?

• yes
• no

Multiple Choice

Pick the ordered pair that is NOT a solution

• (0,0)
• (1,-1)
• (2,0)
• (4,-4)

Multiple Choice

Consider the function y<2x+3. Which is true?

• The line would be solid with shading above.
• The line would be dashed with shading above.
• The line would be solid with shading below.
• None of these.

Multiple Choice

Match the system to the graph (solution set is on the left)

• y ≤ 3x   and  y ≤ -2x + 4
• y > 3x   and  y > -2x + 4
• y < 3x   and  y > -2x + 3
• y > 3x   and  y < -2x + 4

Multiple Choice

Match the system to the graph (solution set is on the left)

• y ≤ 3x   and  y ≤ -2x + 4
• y > 3x   and  y > -2x + 4
• y < 3x   and  y > -2x + 3
• y > 3x   and  y < -2x + 4

Multiple Choice

Which of the following is not a solution to this system of inequalities?

• (0,-1)
• (0,3)
• (4,0)
• (6,-2)

Multiple Choice

Which point is a solution?

• (0, 6)
• (-3, 4)
• (1, 0)
• (-4, 3)

Multiple Choice

Which of the following is not a solution to this system of inequalities?

• (0,-1)
• (0,3)
• (4,0)
• (6,-2)

Multiple Choice

Which system of inequalities matches the graph?

• y > 2x - 4
  y < 2

• y < 2x - 4
  y < 2

• y > 2x - 4
  y <  2 

• y < 2x - 4
  y <   -2

Multiple Choice

Given: y < 4x + 3

Would you draw a solid or dashed line?

• Solid

• Dashed

Multiple Choice

Given: y < 4x + 3

Would shade above or below the line?

• Above

• Below

Multiple Choice

Which system of inequality is shown?

• y > -1/5x - 1/2  
   y ≥ -2x - 4
• y > -1/5x - 1/2  
   y ≤ -2x - 4
• y ≥ -1/5x - 1/2
   y ≤ -2x - 4
• y < -1/5x - 1/2  
  y ≥ -2x - 4

Multiple Choice

Solve the system of inequalities by graphing.

Multiple Choice

Will my line be solid?
5 ≥ x

• Yes
• No

Multiple Choice

Which point is in the overlap (solution) of the shaded region?

• (5,0)

• (-1,8)

• (1,15)

• (10,5)

Multiple Select

Pick ALL correct answers.

Which are solutions to the system?

• (-5,4)

• (-5,-2)

• (0,10)

• (5,10)

• (0,-5)

Multiple Choice

The graph shows which system?

• 
  

  $$y>\frac{2}{3}x+1$$
  

  
  $$y\le-5x+10$$
  

• $$y<\frac{2}{3}x+1$$
  $$y\ge-5x+10$$
  

• $$y<\frac{2}{3}x+1$$
  $$y\le-5x+10$$
  

• $$y>\frac{2}{3}x+1$$
  $$y\ge-5x+10$$
  

Multiple Choice

Which system is graphed?

• 
  

  $$y>-\frac{1}{4}x-3$$  

  
  $$y>2x+6$$  
  

• $$y>\frac{1}{4}x-3$$  
  $$y\le-2x+6$$  
  

• $$y\le-2x+6$$  
  $$y<-\frac{1}{4}x-3$$  
  

• $$y\le-2x+6$$  
  $$y>-\frac{1}{4}x-3$$  
  

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