Linear Equations and Inequalities on the SAT

Why This Topic Matters

Linear equations and inequalities appear in roughly 30-35% of SAT Math questions. Mastering this topic is the single highest-impact thing you can do to improve your math score.

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What Is a Linear Equation?

A linear equation is any equation where the variable has an exponent of 1 (no $x^2$, $\sqrt{x}$, etc.).

Standard form: $Ax + By = C$

Slope-intercept form: $y = mx + b$

Point-slope form: $y - y_1 = m(x - x_1)$

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Solving One-Step Equations

Use inverse operations to isolate the variable.

| Operation in Equation | Inverse Operation | |---|---| | Addition ($+$) | Subtraction ($-$) | | Subtraction ($-$) | Addition ($+$) | | Multiplication ($\times$) | Division ($\div$) | | Division ($\div$) | Multiplication ($\times$) |

Example: $x + 7 = 12$ $x = 12 - 7 = 5$

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Solving Two-Step Equations

Strategy: Undo addition/subtraction first, then undo multiplication/division.

Example: $3x - 5 = 16$

Step 1: Add 5 to both sides: $3x = 21$

Step 2: Divide both sides by 3: $x = 7$

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Multi-Step Equations

When equations have variables on both sides, parentheses, or fractions:

Step-by-Step Strategy

1. Distribute any parentheses
2. Combine like terms on each side
3. Move variables to one side
4. Move constants to the other side
5. Divide to solve

Example: $4(2x - 3) = 5x + 6$ $8x - 12 = 5x + 6$ $3x = 18$ $x = 6$

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Clearing Fractions

Multiply every term by the LCD (least common denominator).

Example: $\frac{x}{3} + \frac{x}{4} = 7$

LCD = 12, so multiply everything by 12: $4x + 3x = 84$ $7x = 84$ $x = 12$

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Special Cases

No Solution

When simplifying leads to a false statement like $5 = 3$, there is no solution.

Example: $2(x + 1) = 2x + 5$ $2x + 2 = 2x + 5$ $2 = 5 \quad \text{(FALSE — no solution)}$

Infinitely Many Solutions

When simplifying leads to a true statement like $3 = 3$, there are infinitely many solutions.

Example: $3(x - 2) = 3x - 6$ $3x - 6 = 3x - 6$ $0 = 0 \quad \text{(TRUE — infinite solutions)}$

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Linear Inequalities

Inequalities use $<$, $>$, $\leq$, $\geq$ instead of $=$.

The One Critical Rule

When you multiply or divide by a negative number, FLIP the inequality sign.

Example: $-3x > 12$ $x < -4 \quad \text{(sign flipped!)}$

Compound Inequalities

$-2 < 3x + 1 \leq 10$ Subtract 1 from all parts: $-3 < 3x \leq 9$ Divide all parts by 3: $-1 < x \leq 3$

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SAT-Specific Question Types

Type 1: "What is the value of x?"

Straightforward solve — isolate the variable.

Type 2: "What is the value of an expression?"

Don't solve for x! Manipulate the equation to find the expression directly.

Example: If $3x + 5 = 17$, what is $6x + 10$? $6x + 10 = 2(3x + 5) = 2(17) = 34$

Type 3: "Which value is NOT a solution?"

Test each answer choice — the one that makes the inequality false is your answer.

Type 4: "For what value of $a$ does the equation have no solution?"

Set up the equation so the variable terms cancel and constants don't match.

Type 5: "For what value of $a$ does the equation have infinitely many solutions?"

Set up the equation so both sides are completely identical.

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Word Problem Translation

| English Phrase | Math Symbol | |---|---| | "is," "equals," "the result is" | $=$ | | "more than," "increased by," "sum" | $+$ | | "less than," "decreased by," "difference" | $-$ | | "times," "product," "of" | $\times$ | | "per," "quotient," "divided by" | $\div$ | | "at least" | $\geq$ | | "at most" | $\leq$ | | "more than" (comparison) | $>$ | | "fewer than" (comparison) | $<$ |

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Common SAT Mistakes

1. Forgetting to distribute the negative sign: $-(x - 3) = -x + 3$, NOT $-x - 3$
2. Not flipping the inequality sign when dividing by a negative
3. Solving for $x$ when the question asks for an expression like $2x + 1$
4. Arithmetic errors with fractions — always find the LCD first
5. Misreading "less than" — "$5$ less than $x$" means $x - 5$, NOT $5 - x$

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Quick Reference: 5-Step SAT Strategy

1. Read the full question — what exactly are they asking for?
2. Simplify — distribute, combine like terms
3. Isolate — get the variable (or expression) alone
4. Check — plug your answer back in
5. Match — make sure your answer matches what was asked

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Practice Tips

• On the SAT, you see roughly 6-8 linear equation questions
• Most can be solved in under 60 seconds with practice
• Watch for "shortcut" questions where you find an expression, not $x$
• If stuck, try plugging in answer choices (backsolving)