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# Expressions, equations, & inequalities FAQ

## How do we collect like terms with integer coefficients?

The commutative property of addition tells us that we can rearrange the order of the terms we're adding together without changing the sum. This is really helpful when we want to combine like terms in an algebraic expression that also has unlike terms.
For example, let's look at this expression:
3, x, plus, 2, y, minus, 5, x, minus, 6, y
Luckily, we can rewrite any subtraction as adding a negative number. So 3, x, plus, 2, y, minus, 5, x, minus, 6, y is the same as 3, x, plus, 2, y, plus, left parenthesis, minus, 5, x, right parenthesis, plus, left parenthesis, minus, 6, y, right parenthesis, and we can use the commutative property of addition on it. That lets us rearrange the terms so that the like terms are next to each other:
3, x, minus, 5, x, plus, 2, y, minus, 6, y
Now we can combine the two x terms and the two y terms.
minus, 2, x, minus, 4, y
And we've simplified the expression!

## How does the distributive property work with variables?

The distributive property works the same way with negative numbers and variables as it does with positive numbers. It allows us to multiply a factor across a set of terms in parentheses.
When it comes to the order of operations, the distributive property can be really helpful. For example, let's say we have the following expression: 2, minus, 3, left parenthesis, x, plus, 4, right parenthesis. Following the traditional order of operations, we would be stuck, because we cannot add x, plus, 4 until we know the value of x. The distributive property lets us multiply the minus, 3 factor by each term inside the parentheses.
\begin{aligned} 2-3(x+4) &= 2+(-3)(x+4)\\\\ &= 2+(-3)(x)+(-3)(4)\\\\ &=2-3x-12 \end{aligned}
Notice that we needed to distribute minus, 3, not just 3. Now we're ready to collect like terms.
2, minus, 3, x, minus, 12, equals, minus, 3, x, minus, 10
So the distributive property allowed us to simplify the expression despite the variable inside the parentheses.

## How do the parts of a linear expression relate to the context it represents?

A linear expression is built out of three main parts: the variable, the coefficient, and the constant. When we use a linear expression to represent a real-world situation, each part of the expression has a different meaning.
The variable is the part of the expression that can change. For example, in the expression 19, minus, 2, x, the variable is x. In a real-world situation, we might use this expression to represent the total length, in centimeters, of a pencil after sharpening it for x minutes if the pencil is originally 19 centimeters long and the sharpener shortens the pencil by 2 centimeters per minute.
The coefficient is the number in front of the variable. In the expression 19, minus, 2, x, the coefficient is minus, 2. In the example about sharpening pencils, this tells us that each minute makes the pencil 2 centimeters shorter.
The constant is the number that doesn't change, no matter what the variable is. In the expression 19, minus, 2, x, the constant is 19. In the example about sharpening pencils, the constant is the pencil's starting 19 centimeter length.
So when we use a linear expression to represent a real-world situation, it's important to pay attention to the different parts of the expression and what they mean in that context.

## How do we solve two-step equations?

A two-step equation is an equation with a variable on one side of the equation and 2 operations. Generally speaking, we solve an equation by working backwards, undoing the operations that have happened to the variable. In that case, we undo the operations in the opposite sequence from the order of operations.
For example, to solve 8, equals, 0, point, 75, b, minus, 1, let's think about the order of operations we would follow to evaluate 0, point, 75, b, minus, 1.
1. Multiply 0, point, 75 times the value of b.
2. Subtract 1.
To solve the equation, we reverse that process.
1. Add 1 to both sides of the equation.
2. Divide both sides of the equation by 0, point, 75.
Let's try it.
\begin{aligned} 8&=0.75b-1\\\\ 8+1&=0.75b-1+1\\\\ 9&=0.75b\\\\ \dfrac{9}{0.75}&=\dfrac{0.75b}{0.75}\\\\ 12&=b \end{aligned}
What would we do if the equation included parentheses? Suppose we have the equation start fraction, 5, divided by, 7, end fraction, left parenthesis, w, plus, 11, right parenthesis, equals, 5. Here are the steps we would take to evaluate start fraction, 5, divided by, 7, end fraction, left parenthesis, w, plus, 11, right parenthesis.
1. Add w and 11 because they are inside parentheses.
2. Multiply by start fraction, 5, divided by, 7, end fraction.
To solve the equation, we reverse that process.
1. Divide by start fraction, 5, divided by, 7, end fraction both sides of the equation by (which is the same as multiplying by start fraction, 7, divided by, 5, end fraction).
2. Subtract 11.
Let's try it.
\begin{aligned} \dfrac{5}{7}(w+11)&=5\\\\ \dfrac{7}{5}\cdot\dfrac{5}{7}(w+11)&=\dfrac{7}{5}\cdot 5\\\\ w+11 &= 7\\\\ w+11-11 &= 7-11\\\\ w &=-4 \end{aligned}
Of course, math is flexible and beautiful. Using the order of operations is only one of many ways to solve. We could also have applied the distributive property to start fraction, 5, divided by, 7, end fraction, left parenthesis, w, plus, 11, right parenthesis, then solved the equation without any parentheses.
Try it yourself with our Two-step equations exercise.

## How do we solve inequalities?

We solve inequalities in the same way that we solve two-step equations, except that we need to remember one rule: if we multiply or divide both sides by a negative number, we need to reverse the inequality sign.
Let's see why this makes sense. We know that 5, is less than, 8. Let's multiply both sides of the equation by minus, 1. Is minus, 5 less than minus, 8? No! minus, 5, is greater than, minus, 8 because minus, 5 is farther to the right on a number line. So when we multiply or divide by a negative number, it swaps the directions of the values on the number line.
For example, to solve minus, 2, x, plus, 1, is greater than, 7, we first subtract 1 from both sides and then divide both sides by minus, 2, which means we need to reverse the inequality sign. The solution is x, is less than, minus, 3.
Try it yourself with our Two-step inequalities exercise.