ACT® · Math: Integrating Essential Skills
Writing and Solving Simple Equations and Inequalities
Writing an equation means turning words into math with a variable standing in for the unknown; solving it means doing the same operation to both sides until that variable sits alone. Inequalities work the same way, except the answer is a whole range of values, not one number.
The hard part of these problems is rarely the algebra. It's reading the sentence and deciding what the variable means, then writing an equation that says exactly what the words say. Once you have something like 3(x - 2) = 5x + 8, the solving is mechanical: distribute, combine like terms, move all the x terms to one side and all the plain numbers to the other, then divide. Every move has to happen on both sides of the equal sign at once, because an equation is a balance. Add 6 to the left and you must add 6 to the right, or the two sides stop matching.
Inequalities use the same moves with one extra rule: multiplying or dividing both sides by a negative number flips the inequality sign. That's because order reverses when you flip the sign of both numbers — 2 < 5, but -2 > -5. Everything else (distributing, combining terms, isolating x) works exactly like an equation.
The payoff for inequalities is different, too. An equation usually pins down one exact value of x. An inequality describes a boundary point and then claims an entire side of the number line beyond it. That's why you write x > 4 instead of a single number: infinitely many values make the statement true, not just one. This connects directly to the rate, percent and multi-step word problems nearby on the test — they all start the same way, by naming a variable and writing down, piece by piece, what the sentence is actually saying.
Key terms
- Variable
- A letter standing in for an unknown number you're trying to find.
- Like terms
- Terms with the same variable part, which can be combined by adding or subtracting their coefficients.
- Isolate the variable
- Use operations on both sides of an equation or inequality until the variable stands alone on one side.
- Boundary value
- The number where an inequality's solution set starts or ends, separating true values from false ones.
- Flip the sign
- The rule that reverses an inequality's direction whenever you multiply or divide both sides by a negative number.
Watch it explained
- How it worksHow does doing the same thing to both sides keep an equation balanced while isolating the variable?
- Worked exampleWalk me through solving 3(x - 2) = 5x + 8 for x the way the ACT wants it done in 40 seconds.
- Worked exampleFind every value of x that satisfies -2x + 7 greater than 15.
- Common mistakeWhy isn't the solution to an inequality a single number the way an equation's solution is?
Check yourself
End of topic.