# Lesson 15

All, Some, or No Solutions

### Problem 1

For each equation, decide if it is always true or never true.

1. $$x - 13 = x + 1$$

2. $$x+\frac{1}{2} = x - \frac{1}{2}$$

3. $$2(x + 3) = 5x + 6 - 3x$$

4. $$x - 3 = 2x - 3 -x$$

5. $$3(x-5) = 2(x-5) + x$$

### Problem 2

Mai says that the equation $$2x + 2 = x +1$$ has no solution because the left hand side is double the right hand side. Do you agree with Mai? Explain your reasoning.

### Problem 3

1. Write the other side of this equation so it's true for all values of $$x$$: $$\frac12(6x-10) - x =$$

2. Write the other side of this equation so it's true for no values of $$x$$: $$\frac12(6x-10) - x =$$

### Problem 4

Here is an equation that is true for all values of $$x$$: $$5(x+2) = 5x+10$$. Elena saw this equation and says she can tell $$20(x+2)+31=4(5x+10)+31$$ is also true for any value of $$x$$. How can she tell? Explain your reasoning.

### Problem 5

Elena and Lin are trying to solve $$\frac12x+3=\frac72x+5$$. Describe the change they each make to each side of the equation.

1. Elena’s first step is to write $$3=\frac72x-\frac12x+5$$.
2. Lin’s first step is to write $$x+6=7x+10$$.

### Solution

(From Unit 4, Lesson 13.)

### Problem 6

Solve each equation and check your solution.

$$3x-6=4(2-3x)-8x$$

$$\frac12z+6=\frac32(z+6)$$

$$9-7w=8w+8$$