Simplifying Expressions versus Solving Equations

Let's look at a typical linear equation that contains some parentheticals.

• Solve 3 + 2[4x − (4 + 3x)] = −1

It is not required that you write out this many steps; once you get comfortable with the process, you'll probably do a lot of this in your head. But until you reach that comfort zone, you should write things out at least this clearly and completely.

Always remember, by the way, that you can check your answers in "solving" problems by plugging the numerical answer back in to the original equation. In this case, the variable is only in terms on the left-hand side (LHS) of the equation; my "check" (that is, my evaluation at the solution value) looks like this:

Since this is what I was supposed to get for the right-hand side (that is, I've shown that the LHS is equal to the RHS), my solution value was correct.

---

Note the difference between the exercise at the top of this page and all the preceeding exercises. This was a "solving" problem, whereas the previous ones had been "simplifying" problems. That is, the exercise above started with an equation (that is, it started something with an "equals" sign in it) and I was supposed to find a solution (that is, I was supposed to find something of the form "(some variable) equals (some number)"). On the other hand, for the previous exercises I had started with an expression (that is, I had started with something containing no "equals" sign) and had ended up with a different version of the same expression (in particular, I ended up with something that still contained no "equals" sign).

Why am I making a big deal about this? Because (warning!) many students, for whatever reason, conflate "equations" and "expressions". That is, given an expression to simplify, the student will somehow turn the problem into an equation to solve. This error will look something like the following:

• Simplify 6 − 4(2x + 3) + 10x

As you can see, the student somehow (at the red arrow) converted the expression "2x − 6" into the equation "2x = 6", and then somehow "solved" the expression. But the original problem didn't have an "equals" sign in it, so there was nothing to solve!

Be careful that you don't do this. And, to help yourself be careful, strive to use "equals" signs only when you're working with equations. If what they gave you didn't start with an "equals" sign in it, then your answer probably shouldn't have one, either.

---

Let's look at some more examples:

• Simplify 6 − 10[x + (2x − 3)] + 12x

---

What if they gave me something that was almost the same as the "simplify the expression" exercise above, but they'd turned it into an equation? Then I'd have something with an "equals" sign in it, and they'd be telling me to "solve", like this:

• Solve 6 − 10[x + (2x − 3)] = −12x

---

By the way, if you take the expression in the previous exercise and set it equal to zero (thus creating an equation which can be solved), you'll get a solution of x = 2.

This is not a mere coincidence. Later, when you study about the real roots of polynomials and how they relate to factorizations, you'll learn more about this. For now, just make sure that you're keeping straight the differences between simplifying expressions and solving equations.

---

• Solve 2(x + 3) = 4 − (2 − x)

---

• Simplify 2(x + 3) − 4 + (2 − x)

---

Let me stress once again: To do "simplification" problems successfully, you need to take the time to write out each step; you must work from the inside out, being especially careful with the "minus" signs when they're right in front of grouping symbols; don't forget the Order of Operations (you'll be needing this for the rest of your life); and don't make the mistake of confusing "simplifying" an expression with "solving" an equation.