Linear Equations & Inequalities — Core Skills

Solving for $x$: The Basics

Part 1 of 2 — The Basics

An equation is two things that are equal, with an equals sign between them. A variable is a letter that stands for a number you do not know yet. Usually that letter is $x$.

Solving an equation means getting $x$ alone on one side. When $x$ is alone, you can read the answer.

The one rule

Whatever you do to one side, do the exact same thing to the other side. That keeps the two sides equal.

To get $x$ alone, you undo what is being done to it:

• If a number is added to $x$, subtract that number from both sides.
• If a number is subtracted from $x$, add that number to both sides.
• If $x$ is multiplied by a number, divide both sides by that number.
• If $x$ is divided by a number, multiply both sides by that number.

Worked example

Solve $3x + 4 = 19$.

Step 1. The $4$ is being added. Undo it by subtracting $4$ from both sides.

$3x + 4 - 4 = 19 - 4$

$3x = 15$

Step 2. The $3$ is multiplying $x$. Undo it by dividing both sides by $3$.

$\frac{3x}{3} = \frac{15}{3}$

$x = 5$

Step 3 (the check). Put $5$ back into the original equation: $3(5) + 4 = 15 + 4 = 19$. That matches, so $x = 5$ is correct.

Checking your answer takes about ten seconds and it tells you for sure whether you are right. Do it every time.

Solving for $x$: Practice

Part 2 of 2 — Practice

The steps, every time

1. Look at what is happening to $x$.
2. Undo any adding or subtracting first. Do it to both sides.
3. Undo any multiplying or dividing second. Do it to both sides.
4. Put your answer back into the original equation to check it.

Inequalities work almost the same way

An inequality uses one of these signs instead of an equals sign:

• $>$ means "greater than"
• $<$ means "less than"
• $\geq$ means "greater than or equal to"
• $\leq$ means "less than or equal to"

You solve an inequality with the exact same steps. There is one extra rule:

If you multiply or divide both sides by a negative number, flip the direction of the sign.

Worked example. Solve $-2x < 8$.

Divide both sides by $-2$. Because $-2$ is negative, the $<$ flips to $>$.

$x > \frac{8}{-2}$

$x > -4$

That is the whole trick. Nothing else about inequalities is different.