3. Algebraic Expressions, Equations & Inequalities

This is the section where letters show up and a lot of students panic. Don't. A letter is just a number you haven't met yet. Almost every problem here comes down to two skills: cleaning up a messy expression, and undoing an equation one step at a time until $x$ is standing all by itself. The ISEE isn't trying to test whether you're a genius. It's testing whether you're careful with minus signs. So let's be careful with minus signs.

The practice is mixed into the lesson: each question sits right after the idea it needs, and the Mixed Practice at the bottom takes that hint away. Flip the switch above any question to type your answer instead. Numbers and fractions work (17/4), and so do expressions (5x + 2, 3(2x + 3)), equations for a letter (y = -2x + 7), inequalities (x > 2, -4 <= x <= 10, x < -3 or x > 5), and the words "no solution" or "infinitely many." A few versions need their answer choices, and they'll tell you so.

Expressions, Terms & Like Terms

First, the vocabulary. You're going to see the words "expression" and "equation" all over the test, and they are not the same thing. But before either one, meet the letters.

Mix variables and numbers together with some operations, and you've built an expression.

So $3x-7$, $2(x+5)$, and $\dfrac{x}{3}+4$ are all expressions. No equals sign means there's nothing to solve. All you can do with an expression is simplify it or plug numbers into it. To simplify, you need to know what the pieces are called.

Read that last part again: the minus sign belongs to the term that follows. That one sentence is responsible for about half the wrong answers in this whole section. In $7x-3x+5$, the middle term isn't $3x$. It's $-3x$. The minus sign is glued to it and goes wherever it goes.

Every sign in front of a term is part of that term. Keep that in mind and the rest of this page gets a lot easier. The number stuck to the front of a term gets a name too.

Here's all of that new vocabulary on one expression.

And when two terms have exactly the same letters, they're teammates.

Here's what that looks like on a messy expression, with each kind of term in its own color.

Think of it like sorting laundry. Socks go with socks, shirts go with shirts. $x$'s go with $x$'s, $x^{2}$'s go with $x^{2}$'s, and plain numbers go with plain numbers. You would never fold a sock into a shirt, and you never add $3x^{2}$ to $2x$. Try it yourself below.

Notice what the bins are really doing. Each one just adds up the numbers in front of its terms. That's the whole rule:

Here it is on a real expression, minus signs and all.

Here's the same idea with tiles you can actually count. A positive tile and a negative tile of the same kind make a zero pair: together they're worth $0$, so they cancel. Whatever survives is the simplified expression.

Now test your eye. Some of these really are the same expression, and some only look like it.

Your turn. Sort the terms in your head, give every term its own sign, and add up each pile.

The Distributive Property

Parentheses are the other thing standing between you and a simplified expression. The number out front has to multiply everything inside, not just the first thing it bumps into.

Same rule, two directions. Distributing opens the parentheses up; factoring packs them back in. Why does the number out front hit both terms? Picture it as the height of a rectangle. The area is both pieces, not just the first one.

Now a real one, with two sets of parentheses to open.

Now the sneaky one. A minus sign in front of parentheses is really a $-1$ being distributed, and it flips the sign of every term inside.

That one is easy to understand and easy to forget in the middle of a test. So here's a habit.

Watch the number out front hit every term, and then watch what happens when it doesn't.

Put all of it together and you get the full warm-up: distributing, minus signs, and like terms in one problem.

Your turn: distribute first, then combine like terms.

Factoring runs the same rule backward. Find the biggest number that divides every term, and pull it out front.

Your turn. Pull out the greatest common factor, and check by distributing back.

The last thing you can do with an expression is plug numbers in. This is where negative numbers love to cause trouble.

Your turn. Parentheses on first, then the exponent, then everything else.

Solving Equations

Now we add an equals sign, and suddenly there's something to find.

The best way to picture an equation is a balance scale. The left side and the right side weigh exactly the same. You're allowed to do anything you want to the scale (add weight, take weight off, cut everything in half) as long as you do it to both sides, because then it stays balanced. Your mission is to keep doing that until one pan holds a single $x$ and nothing else. Whatever's on the other pan is the answer.

"Whatever you do to one side, you must do to the other." That's the only rule. The order that works best: get all the $x$'s on one side, then get the numbers off that side, then divide.

One version of this looks scarier than it is.

Your turn: $x$'s to one side, numbers to the other, then divide.

Why undo things in that order? Think of $x$ as a present that got wrapped: the last layer on is the first layer off.

So in $4x+7=31$ the $+\,7$ comes off first, and the $\times\, 4$ comes off last. Try a few two-step equations.

If there are parentheses, distribute first. Then it's the same three steps.

And here's our old friend, the minus sign in front of parentheses.

Your turn. Clear the parentheses first, minus signs included.

Word problems are just equations wearing a disguise. Translate one phrase at a time.

One phrase in particular trips everyone up, because it reads backwards.

Translating is a skill all by itself, so practice it on its own first. Build each sentence's equation from the tiles, one phrase at a time.

And here's the order trap by itself. Each phrase means either $2n-7$ or $7-2n$. Which one?

Now translate and solve.

Fractions look scary, but there's one move that makes them vanish: multiply every single term on both sides by a number that all the denominators go into.

Here's that one written out, with a check at the end.

Same trick when there's a fraction on each side.

Notice that the $2$ had to hit the whole numerator. That's worth a reminder.

Your turn: clear the fractions, then solve.

Sometimes an equation has more than one letter in it, and the question only wants one of them.

Same balance-scale moves as always. The other letters just come along for the ride.

That last step, multiplying by $-1$, is exactly where students slip.

Formulas are literal equations too, and they work exactly the same way.

Try it with moves of your own. Each one happens to both sides.

Your turn. Get the letter they ask for alone.

Other times there are too many letters to solve for each one, and you don't need to.

Here's that move in slow motion: spot the chunk, then swap in its value.

So the habit is simple.

Your turn: find the chunk, then use it.

One more trick: if the same chunk shows up on both sides, pull it out and cancel it.

That "$x \neq 1$" is doing real work: you can't divide by a chunk that might be zero. Your turn.

Identities & No Solution

Most equations have exactly one answer. But sometimes, while you're solving, every $x$ disappears. When that happens, look at what's left over.

And the exact opposite can happen too.

Here's one of each.

Load both of these into the Balance Scale above. With $2(x+3)=2x+6$, the two pans are identical, so they balance no matter what $x$ is. With $x=x+1$, the right pan is always heavier by exactly $1$, so the scale can never balance.

Here's another way to see all three cases. Make each side of the equation a bar, and slide $x$ around looking for a spot where the bars match.

So here's the rule of thumb:

• The $x$'s cancel and you're left with something true (like $6=6$): infinitely many solutions.
• The $x$'s cancel and you're left with something false (like $0=1$): no solution.
• The $x$'s don't cancel: exactly one solution.

Here's the rule in action on a typical ISEE question.

Fractions don't change anything: clear them first, then look at what's left.

Your turn to sort them. Simplify each one in your head first.

Now on your own. Some of these ask which equation, and some ask you to solve one.

Inequalities

An equation says two things are equal. An inequality says one is bigger.

The good news: you solve inequalities exactly the way you solve equations. Same steps, same order. The bad news: there's one rule that equations don't have, and the ISEE tests it constantly.

Why? Because negatives reverse the order of numbers. $2<5$, but $-2>-5$ (negative two is warmer than negative five). Multiply by a negative and the bigger side becomes the smaller side, so the sign has to turn around. Slide the multiplier and watch it happen.

Here's the flip in a real problem.

Not every minus sign means a flip, though. Only multiplying or dividing by a negative does.

And fractions? Clear them first. Multiplying by a positive number never flips anything.

Your turn. Watch for the flip, and only flip when it's earned.

Here's the best safety net for every one of those.

Try it: predict which side gets shaded, then let a test point check you.

Testing a single number is a question on its own, too. Which of these numbers actually works?

Sometimes $x$ is trapped between two numbers at once.

Solve it the same way, but every move has to happen to all three parts.

Here's that squeeze on a number line: adding slides the range, dividing shrinks it.

Dividing by a negative works too, with one twist.

The test asks this both ways: solve and pick the matching number line, or read a number line and pick the matching inequality. Either way, check the dots (open or closed) and whether the shading is between the ends or outside them.

Your turn. Every move, all three parts.

Absolute Value

Absolute value has a reputation for being confusing, but it really comes down to one word: distance.

So $|5|=5$ and $|-5|=5$: both numbers are $5$ steps from $0$. When there's arithmetic inside the bars, do it first, then take the distance.

Your turn. Inside the bars first, then the distance.

So $|x-3|$ is the distance from $x$ to $3$. Saying $|x-3|=7$ is saying "$x$ is exactly $7$ away from $3$," and there are two numbers like that: one $7$ to the right, one $7$ to the left. That's why absolute value equations usually have two answers.

Watch it happen on a number line, plus sign inside and all.

Here's the same equation as a test question.

Your turn. Find the center first.

The same distance idea handles inequalities. "At most $7$ away" is everything squeezed in the middle. "At least $7$ away" is everything out past the edges, heading off in both directions.

Play with it below. The caption writes out the inequality for you, and the little test pointer underneath checks any number you like.

Here are the two shapes on paper. First, "at most": the middle.

Then "more than": the two pieces out past the edges.

Your turn. Middle or edges?

Two answers is the usual case for an equation, but not the only one. Predict before you peek.

When there's an $x$ on both sides, you can't just use the distance picture. Split it into two plain equations instead: the inside equals the right side, or the inside equals the negative of the right side.

Both of those survived the check. They don't always.

And one more habit for Case 2.

Your turn to be the judge: keep each candidate, or throw it out?

Now solve some all the way through: two cases, then the check.

Column A vs. Column B

The ISEE loves quantitative comparison, and with algebra it has a favorite trick: give you an $x$ with no value and hope you panic and pick (D), "cannot be determined." Don't. Simplify first. Lots of the time the $x$'s cancel and the answer is settled no matter what $x$ is.

Don't believe it? Watch the two columns while $x$ moves around.

When they do give you a value, plug it in carefully, especially inside absolute value bars.

And sometimes each column is its own little equation to solve.

One last rule of thumb, for choice (D).

Mixed Practice

Every question so far came right after its idea, which is a hint the real test never gives. Down here the hint is gone. First, the quantitative comparison: two columns, four fixed choices, and any idea from this section fair game.

Next, the mixed set. Every question type on this page except the quantitative comparison goes into one hat, and each click draws a new one at random. Flip the switch to type a number answer; answers that are expressions or inequalities stay multiple choice down here.

Alright, time for something with a little heat. These are absolute value inequalities with an $x$ on both sides. The trick is to trap the inside between the right side and its negative, like $-(x+3)\le 2x-6\le x+3$, then solve each half separately. For "greater than," split it instead: the inside is at least the right side, or at most its negative. Either way, watch your signs when you make the right side negative.