# Unit 4 Family Materials

Inequalities, Expressions, and Equations

### Writing Equivalent Expressions

This week your student will be working with equivalent expressions (expressions that are always equal, for any value of the variable). For example, $$2x+7+4x$$ and $$6x+10-3$$ are equivalent expressions. We can see that these expressions are equal when we try different values for $$x$$.

$$\displaystyle 2x+7+4x$$ $$\displaystyle 6x+10-3$$
when $$x$$ is 5 $$\displaystyle 2\boldcdot5+7+4\boldcdot5$$ $$\displaystyle 10 + 7 + 20$$ $$\displaystyle 37$$ $$\displaystyle 6\boldcdot5+10-3$$ $$\displaystyle 30+10-3$$ $$\displaystyle 37$$
when $$x$$ is -1 $$\displaystyle 2\boldcdot\text-1+7+4\boldcdot\text-1$$ $$\displaystyle \text-2 + 7 + \text-4$$ $$\displaystyle 1$$ $$\displaystyle 6\boldcdot\text-1+10-3$$ $$\displaystyle \text-6+10-3$$ $$\displaystyle 1$$

We can also use properties of operations to see why these expressions have to be equivalent—they are each equivalent to the expression $$6x+7$$.

Match each expression with an equivalent expression from the list below. One expression in the list will be left over.

1. $$5x + 8 - 2x + 1$$
2. $$6(4x - 3)$$
3. $$(5x + 8) - (2x + 1)$$
4. $$\text-12x + 9$$

List:

• $$3x + 7$$
• $$3x + 9$$
• $$\text-3(4x-3)$$
• $$24x + 3$$
• $$24x - 18$$

Solution:

1. $$3x + 9$$ is equivalent to $$5x + 8 - 2x + 1$$, because $$5x+\text-2x=3x$$ and $$8 + 1 = 9$$.
2. $$24x - 18$$ is equivalent to $$6(4x-3)$$, because $$6\boldcdot4x = 24x$$ and $$6\boldcdot\text-3=\text-18$$.
3. $$3x + 7$$ is equivalent to $$(5x + 8) - (2x + 1)$$, because $$5x - 2x = 3x$$ and $$8 - 1 = 7$$.
4. $$\text-3(4x-3)$$ is equivalent to $$\text-12x+9$$, because $$\text-3\boldcdot4x=\text-12x$$ and $$\text-3\boldcdot\text-3=9$$.

### Equations in One Variable

This week your student will work on solving linear equations. We can think of a balanced hanger as a metaphor for an equation. An equation says that the expressions on either side have equal value, just like a balanced hanger has equal weights on either side.

If we have a balanced hanger and add or remove the same amount of weight from each side, the result will still be in balance.

We can do this with equations as well: adding or subtracting the same amount from both sides of an equation keeps the sides equal to each other. For example, if $$4x+20$$ and $$\text-6x +10$$ have equal value, we can write an equation $$4x+20=\text-6x+10$$. We could add -10 to both sides of the equation or divide both sides of the equation by 2 and keep the sides equal to each other. Using these moves in systematic ways, we can find that $$x=\text-1$$ is a solution to this equation.

Elena and Noah work on the equation $$\frac12 \left(x+4\right) = \text-10+2x$$ together. Elena’s solution is $$x=24$$ and Noah’s solution is $$x=\text-8$$. Here is their work:

Elena:

\begin{align} \frac12 \left(x+4\right) &= \text-10+2x\\ x+4 &= \text-20+2x\\ x+24 &= 2x\\ 24&=x\\ x&=24\end{align}

Noah:

\begin{align} \frac12 \left(x+4\right) &= \text-10+2x\\ x+4 &=\text -20+4x\\ \text-3x+4 &= \text-20\\ \text-3x &= \text-24\\ x&=\text-8\end{align}

Do you agree with their solutions? Explain or show your reasoning.

Solution:

No, they both have errors in their solutions.

Elena multiplied both sides of the equation by 2 in her first step, but forgot to multiply the $$2x$$ by the 2. We can also check Elena’s answer by replacing $$x$$ with 24 in the original equation and seeing if the equation is true. $$\displaystyle \frac12 \left(x+4\right) =\text -10+2x$$ $$\displaystyle \frac12 \left(24+4\right) =\text -10+2(24)$$ $$\displaystyle \frac12 \left(28\right) = \text-10+48$$ $$\displaystyle 14=38$$ Since 14 is not equal to 38, Elena’s answer is not correct.

Noah divided both sides by -3 in his last step, but wrote -8 instead of $$8$$ for $$\text-24 \div \text-3$$. We can also check Noah’s answer by replacing $$x$$ with -8 in the original equation and seeing if the equation is true. Noah’s answer is not correct.