Algebra I

Creating Expressions, Equations, and Inequalities

One of the most critical skills in algebra is learning to write an equation.  Being able to translate a problem -- whether a word problem or a real life problem -- into an equation opens up an entire realm of math.

So, how do you translate words into an equation?

First, some basics:

Create Equations from Word Problems

One of the algebra skills that students struggle with the most is writing equations from word problems. Ironically, translating real life problems into math is one of the key ways that algebra can become useful in real life (All those times you ask yourself, why do I need to learn this?  This is why you need to learn algebra!).

Interpret Equations

Equations are mathematical sentences.  We write equations to solve for variables that we don't know, but can predict based upon other variables. Some of the most useful -- and difficult -- math problems ask students to write or interpret equations.  What does this variable mean?  What happens to variable $n$ when variable $x$ goes up or down?

Absolute Value Inequalities

Just as there are absolute value equations, there are also absolute value inequalities.  You solve absolute value inequalities in the exact same way that you deal with absolute value equations, but with the twist that is similar to the twist involved with regular inequalities: you have to flip the inequality sign when you flip the signs of the numbers.

To solve an absolute value inequality:

Absolute Value (in equations with extra terms)

You know that, in order to solve for a variable in an absolute value equation, you set the equation inside the absolute value to both the positive and negative versions of the answer.

What do you do when there are additional terms in the equation, outside of the absolute value expression?

Whenever an absolute value equation has additional terms, outside of the absolute value expression, you use algebra to get rid of those extra terms BEFORE you set the equation equal to both the positive and negative forms of the answer.

Absolute Value (in equations)

You have already learned that the absolute value of a variable tells you the value of the variable, but not whether the variable is positive or negative (and, without more information, you have to assume that it could be either).

Any time you solve for a variable whose absolute value is given, your answer will be two possible answers, as in the following equation (to review, go to the Absolute Value (with variables) lesson).

Absolute Value (with variables)

The absolute value of a number is always positive.  It represents the number's distance from 0, and distance is always positive.

However, when you are working with variables and absolute value, you have to remember that while the absolute value of a number is always positive, the original number may have been negative.

$\mid 6 \mid = 6$

So, if you are given $\mid x \mid = 6$, then $x$ might equal $6$.

But, remeber that:

$\mid ^-6 \mid = 6$

Absolute Value

The absolute value of a number is the distance that number is from 0.

That means that 3 has an absolute value of 3. As you can see from the number line below, 3 is three units away from 0.

Solve and Substitute (Substitution)

It's easy to solve an equation with one variable: you isolate the variable and find what number it is equal to.  But, when you have equations with two variables, it can be impossible to find a single numerical value for either variable.  However, if you have two equations, with the same variables, you can often solve for a variable in one equation and then substitute that answer into the other equation to solve.  Remember, in math, you can always substitute variables, terms, and numbers for each other if they are equal.

Create Systems of Equations from Word Problems

Some word problems are best solved by creating a system of equations (or two equations that use the same variables).

How do you identify those word problems?

Word problems best solved with a system of equation usually give two different totals. One total is typically a straight sum (e.g., adult tickets plus kid tickets equal total tickets) and the other is a sum that uses a multiplier (e.g., adult tickets, which cost \$10, plus kid tickets, which cost \$6 equal sum total cost in dollars).