6.2.6 Possible Solutions

Is there always a solution?

A case for infinite or no solution

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In this lesson, you will continue to practice your strategies of combining like terms, removing zeros, and balancing to simplify and compare two expressions. You will also encounter unusual situations where the solution may be unexpected. As you work today, focus with your team on these questions:

What if both sides are not equal?

Are there many values of

that will make the expressions equal?

Is there always a solution?

Possible Solutions

Some situations have one solution, others have no solution, and still others have all numbers as solutions. When simplified if the equation can be simplified to:

x = x then the equation has infinite solutions because it doesn't matter what the value of x is it is true.

2x = x then the equation has only one solution and that solution is 0 because that is the only way that the equation can be true.

x + 1 = x - 1 then the equation has no solution because there is no solution that would make the equation true.

x = a constant then that constant is the one and only possible solution to make the equation true.

Multiple Choice

How many solutions does the following equation have?

x = x

• One Solution

• Infinite Solutions

• No Solutions

Multiple Choice

How many solutions does the following equation have?

x + 1 = x

• One Solution

• Infinite Solutions

• No Solutions

Multiple Choice

How many solutions does the following equation have?

x = 2x

• One Solution

• Infinite Solutions

• No Solutions

Multiple Choice

How many solutions does the following equation have?

x + x = 2 + x

• One Solution

• Infinite Solutions

• No Solutions

Multiple Choice

How many solutions does the following equation have?

x + x = x - x

• One Solution

• Infinite Solutions

• No Solutions

Multiple Choice

How many solutions does the following equation have?

x + x = 2x

• One Solution

• Infinite Solutions

• No Solutions

Multiple Choice

How many solutions does the following equation have?

$$\times\cdot\times=\times^2$$

• One Solution

• Infinite Solutions

• No Solutions

Multiple Choice

How many solutions does the following equation have?

x - 1 = x

• One Solution

• Infinite Solutions

• No Solutions

Multiple Choice

Kelly is 6 years younger than her twin brothers Bailey and Larry.  How old will Kelly be when the sum of her brothers ages is 12 more than twice Kelly’s? 

Which Equation matches the situation?

• 2(x + 6) = 2x + 12

• 2x + 6 = 2x + 12

• 2(x + 6) = 2x + 6

• 2x + 6 = 2x + 12

Open Ended

What is the solution to the equation:

2(x + 6) = 2x + 12

​One Solution

Equations that have exactly one solution.

Multiple Choice

Solve for x.

• x = 3

• x = 6

• x = -2

• x = -5

Draw

Solve for n. Show all your work.

Multiple Choice

• v = 15

• v = 3

• v = 24

• v = 33

Multiple Choice

• r = 6

• r = 7

• r = -6

• r = 13

2x + 3 = 2x + 7

Multiple Choice

How many x's can we remove from both sides of the equation 2x + 3 = 2x + 7?

• 1 from each side of the equation.

• 4 from each side of the equation.

• 2 from each side of the equation.

• None

Multiple Choice

What is you new equation after removing 2x from each side?

• 2x + 3 = 4x + 2

• 3 = 7

• 2x + 3 = 7

• 3 = 2x + 7

Open Ended

Since we simplify down to 3 = 7 as our equation, what can we say is our solution to the equation? Explain.

​No Solutions

​Equations that have no solutions. No values we assign to the variable will make the equation true.

Multiple Choice

What should be my first step to solve the equation?

• Subtract 2x from both sides.

• Subtract 6 and add 2x to both sides.

• Add 6 to both sides.

• Distribute -2 by multiplying it to x and +3.

What would be our solution? Explain.

Open Ended

What would be our solution? Explain.

​Infinite Solutions

There is an infinite amount of solutions to the equation. Any value ​for the variable would make the equation true.

​Infinite Solutions

There is an infinite amount of solutions to the equation. Any value ​for the variable would make the equation true.

Multiple Choice

How many solutions are there in the following equation?

• one

• no solutions

• infinite solutions

Multiple Choice

How many solutions are there in the following equation?

• one

• no solutions

• infinite solutions

Multiple Choice

How many solutions are there in the following equation?

• one

• no solutions

• Infinite solutions

Open Ended

Describe the differences with equations that have one, none, or infinite solutions?

Multiple Choice

How many solutions does this equation have?  3x + 2 = 3x + 2

• One Solution

• No Solution

• Infinite Solutions

Multiple Choice

How many solutions does this equation have?  3x -20 = 3x + 2

• One Solution

• No Solution

• Infinite Solutions

Multiple Choice

How many solutions does the equation have
6x + 3 = 6x + 3

• One Solution

• No Solution

• Infinite Solutions

Multiple Choice

How many solutions does the equation have
3t + 5 = 4t + 7

• One Solution

• No Solution

• Infinite Solutions

Multiple Choice

How many solutions does the equation have
4 ( 8n - 1 ) = 19 + 32n

• One Solution

• No Solution

• Infinite Solutions

Multiple Choice

How many solutions does the equation have
3x + 7 + 2x = 5x + 7

• One Solution

• No Solution

• Infinite Solutions

Multiple Choice

How many solutions does the equation have
4y + 10 = 10 + 4y

• One Solution

• No Solution

• Infinite Solutions

Multiple Choice

Fill in the blank so that the following equation has NO solution.
__x + 12 = 2x - 7

• 4

• 2

• 9

• 7

Match

Match the following

• 2 = 3

• x = x

• x = 3

No solution

Infinite Solution

On solution

• No solution

• Infinite Solution

• On solution

Review/Preview

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