Equations & Inequalities

An equation is a math sentence that says two things are equal. It always has an equals sign, $=$, with an expression on each side, like $x + 5 = 12$. A letter such as $x$ is a variable, an unknown number we want to find. To solve an equation means to find the value of the variable that makes the sentence true. That value is called the solution.

Every operation has an inverse (an opposite) that undoes it:

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• Addition and subtraction undo each other.
• Multiplication and division undo each other.

To get the variable alone, use the inverse of whatever is happening to it.

A one-step equation needs just a single inverse operation to get the variable alone. Look at what is being done to the variable, then do the opposite to both sides.

A two-step equation has two operations, like $2x + 3 = 11$. Undo them in the reverse of the order of operations: first undo addition or subtraction, then undo multiplication or division. Peel the variable free from the outside in.

Sometimes a side has more than one term with the same variable. Like terms have the same variable, so you can combine them: $3x + 2x = 5x$. Simplify each side first, then solve as a normal equation.

To solve a word problem, let a variable stand for the unknown, translate the words into an equation, then solve. Watch for signal words: “more than” or “increased by” means add; “less than” means subtract; “times” or “product” means multiply; “split” or “per” often means divide.

An inequality compares two expressions that are not necessarily equal. Instead of one solution, it usually has many. The four symbols are:

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• $<$ means “less than”
• $>$ means “greater than”
• $\le$ means “less than or equal to”
• $\ge$ means “greater than or equal to”

For example, $x > 5$ means every number bigger than $5$ is a solution.

You solve inequalities almost exactly like equations: use inverse operations and do the same thing to both sides. One-step and two-step inequalities work the same way. There is just one special rule, coming next.

When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. A “$<$” becomes “$>$,” and a “$\le$” becomes “$\ge$” (and the other way around). This is the single most important rule that makes inequalities different from equations.

Because an inequality has many solutions, we picture them on a number line. Put a circle at the boundary number and shade the direction of all the solutions:

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• Use an open circle (hollow) for $<$ or $>$, because the boundary number is not included.
• Use a closed circle (filled in) for $\le$ or $\ge$, because the boundary number is included.

Then shade to the right for greater-than, or to the left for less-than.

An open circle means the endpoint is not included.

When the variable appears on both sides, like $5x - 3 = 2x + 9$, first gather all the variable terms onto one side and all the plain numbers onto the other. Add or subtract a variable term from both sides to move it, then finish with the usual two-step method. If the equation has fractions, you can clear the denominators first: multiply every term on both sides by the least common denominator (LCD). This turns a messy fraction equation into a clean whole-number one.

Sometimes, while solving, the variable cancels out completely and you are left with a statement of pure numbers. What that statement says tells you the answer:

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• If you get a false statement like $5 = -3$, there is no solution. No number could ever make the original equation true.
• If you get a true statement like $8 = 8$, there are infinitely many solutions. Every number works, because the two sides were really the same expression in disguise (this is called an identity).

For example, $2x + 5 = 2x - 3$ gives $5 = -3$ after subtracting $2x$: no solution. But $4x + 8 = 4(x + 2)$ becomes $4x + 8 = 4x + 8$, then $8 = 8$: infinitely many solutions.

A system is two equations that must both be true at the same time, usually with two variables like $x$ and $y$. The solution is the pair of values that satisfies both. One reliable method is substitution: if one equation already tells you what a variable equals (for example $y = x + 1$), plug that expression into the other equation. This leaves a single equation in one variable, which you already know how to solve. Then back-substitute to find the second variable.

A compound inequality traps the variable between two bounds, like $-1 < 2x + 3 \le 9$. It really means two inequalities joined by “and”: the middle is greater than $-1$ and at most $9$. To solve, do the same operation to all three parts at once. Interval notation is a compact way to write the answer: a round bracket $(\ )$ excludes an endpoint (matches $<$ or $>$), and a square bracket $[\ ]$ includes it (matches $\le$ or $\ge$). So $-2 < x \le 3$ is written $(-2,\ 3]$.