Linear Equations and Inequalities

What Is a Linear Equation?

A linear equation is any equation that can be written in the form $ax + b = c$, where $a$, $b$, and $c$ are constants and $x$ is the variable. The word "linear" means the variable has an exponent of 1 — no $x^2$, no $\sqrt{x}$, no $x$ in a denominator multiplied by another variable. When graphed, a linear equation produces a straight line.

The standard forms you will encounter on the ACT include:

• Slope-intercept form: $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept
• Standard form: $Ax + By = C$, where $A$, $B$, and $C$ are integers
• Point-slope form: $y - y_1 = m(x - x_1)$, which is useful when you know one point and the slope

Each form has its advantages depending on the context. On the ACT, slope-intercept form is the most common because it immediately reveals the slope and the $y$-intercept, making it easy to graph or compare lines.

Solving One-Variable Linear Equations

The goal is to isolate the variable on one side of the equation using inverse operations. Follow this systematic process:

1. Distribute any parentheses: $3(x + 2) = 15$ becomes $3x + 6 = 15$
2. Combine like terms on each side of the equation
3. Add or subtract to move all variable terms to one side and constants to the other
4. Multiply or divide to isolate the variable completely

Detailed example: Solve $4(2x - 3) + 5 = 21$.

$4(2x - 3) + 5 = 21$ $8x - 12 + 5 = 21 \quad \text{(distribute the 4)}$ $8x - 7 = 21 \quad \text{(combine } {-12} \text{ and } {+5}\text{)}$ $8x = 28 \quad \text{(add 7 to both sides)}$ $x = 3.5 \quad \text{(divide both sides by 8)}$

Always verify by substituting back: $4(2(3.5) - 3) + 5 = 4(7 - 3) + 5 = 4(4) + 5 = 16 + 5 = 21$ ✓

Equations with Fractions

When an equation contains fractions, multiply every term by the least common denominator (LCD) to clear the fractions before solving. This eliminates messy fraction arithmetic.

For $\frac{x}{3} + \frac{x}{4} = 7$, the LCD is 12:

$12 \cdot \frac{x}{3} + 12 \cdot \frac{x}{4} = 12 \cdot 7$ $4x + 3x = 84$ $7x = 84$ $x = 12$

Solving Two-Variable Systems of Equations

The ACT frequently asks you to solve a system of two linear equations simultaneously. The two main algebraic methods are:

• Substitution: Solve one equation for a variable, then plug that expression into the other equation. Best when one variable is already isolated or has a coefficient of 1.
• Elimination (addition/subtraction): Multiply one or both equations by constants so that when you add or subtract them, one variable cancels out. Best when neither variable is easily isolated.

Substitution example: Solve $y = 2x + 1$ and $3x + 2y = 16$.

Substitute $y = 2x + 1$ into the second equation: $3x + 2(2x + 1) = 16$ $3x + 4x + 2 = 16$ $7x = 14$ $x = 2, \quad y = 2(2) + 1 = 5$

Elimination example: Solve $2x + y = 10$ and $x - y = 2$.

Add the equations: $(2x + y) + (x - y) = 10 + 2$, giving $3x = 12$, so $x = 4$ and $y = 2$.

Special cases: If the system simplifies to $0 = 0$ (a true statement), the lines are the same (infinitely many solutions). If it simplifies to $0 = 5$ (a false statement), the lines are parallel (no solution).

Linear Inequalities

Linear inequalities work just like linear equations with one critical difference: when you multiply or divide both sides by a negative number, you must flip the inequality sign. This rule is the source of the single most common error students make on inequality problems.

For $-3x + 6 > 12$:

$-3x > 6 \quad \text{(subtract 6 from both sides)}$ $x < -2 \quad \text{(divide by } {-3} \text{ and FLIP the sign)}$

Notice the sign changed from $>$ to $<$ when we divided by $-3$. Forgetting this step is the number-one mistake on inequality questions.

Compound Inequalities

Compound inequalities combine two inequalities into a single statement. For example, $-4 < 2x + 6 \leq 10$ means you solve both parts simultaneously by performing the same operation on all three parts:

$-4 < 2x + 6 \leq 10$ $-10 < 2x \leq 4 \quad \text{(subtract 6 from all parts)}$ $-5 < x \leq 2 \quad \text{(divide all parts by 2)}$

This means $x$ is greater than $-5$ and less than or equal to $2$.

Absolute Value Equations and Inequalities

The ACT occasionally tests absolute value. The absolute value $|a|$ represents the distance of $a$ from zero on the number line, and it is always non-negative.

Key rules:

• $|x| = a$ (where $a \geq 0$) means $x = a$ or $x = -a$
• $|x| < a$ means $-a < x < a$ (the "sandwich" — solutions are between $-a$ and $a$)
• $|x| > a$ means $x > a$ or $x < -a$ (solutions are outside the interval)

Example: Solve $|2x - 5| = 9$.

Case 1: $2x - 5 = 9 \Rightarrow 2x = 14 \Rightarrow x = 7$

Case 2: $2x - 5 = -9 \Rightarrow 2x = -4 \Rightarrow x = -2$

The solution set is $\{-2, 7\}$.

Important: If an absolute value equation equals a negative number (e.g., $|x + 3| = -4$), there is no solution because absolute value cannot be negative.

Translating Word Problems into Equations

Many ACT linear equation questions are disguised as word problems. The key skill is translating English into algebraic notation. Memorize these translations:

English Phrase	Math Symbol
"is," "equals," "was," "will be"	$=$
"more than," "increased by," "added to"	$+$
"less than," "decreased by," "subtracted from"	$-$
"times," "of," "product," "each"	$\times$
"per," "divided by," "ratio of," "quotient"	$\div$
"at least"	$\geq$
"no more than," "at most"	$\leq$
"between"	compound inequality

Tip: Always define your variable first ("Let $x$ = the number of hours"), then translate the problem sentence by sentence.

Tip 1: Plug in the Answer Choices (Backsolving)

When an ACT question asks "What is the value of $x$?" and gives five answer choices (A through E), you can substitute each choice back into the equation to see which one works. Start with choice C (the middle value, since choices are usually in order) — if the result is too big, try a smaller choice; if too small, try larger. This strategy is fast, reliable, and completely eliminates algebra errors.

Tip 2: Pick Numbers for Abstract Problems

When a problem uses variables in the answer choices (e.g., "Which expression represents...?"), assign simple, convenient numbers like $x = 2$ or $n = 3$ to the variables, compute the value of the expression in the problem, then check which answer choice gives the same value. This converts abstract algebra into simple arithmetic.

Tip 3: Watch for "No Solution" and "Infinite Solutions"

If simplifying an equation leads to a false statement like $0 = 5$, there is no solution (the lines are parallel). If it leads to a true statement like $3 = 3$, there are infinitely many solutions (the equations describe the same line). The ACT tests this concept occasionally, and students who have never seen it are caught off guard.

Tip 4: Manage Your Time Aggressively

At roughly one minute per question, you should not spend more than 60–90 seconds on any linear equation problem. These should be quick wins. If you find yourself stuck or confused, circle the question, make your best guess (remember: no penalty for wrong answers!), and move on. You can always come back if time permits.

Tip 5: Always Re-Read the Question

The ACT sometimes asks for an expression like $2x$ or $x + y$ rather than just $x$ alone. A common trap is to solve for $x$, find $x = 4$, and select 4 as the answer — but the question actually asked for $2x$, which is 8. Always re-read the specific question before selecting your answer.

Tip 6: Check Your Work by Substitution

After solving for $x$, take 10 seconds to substitute your answer back into the original equation. This catches arithmetic errors and gives you confidence. On the ACT, certainty on easy questions frees up mental energy for hard ones.