Linear Equations and Inequalities

Q: How can I study Linear Equations and Inequalities effectively?

Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 16 problems provided, checking solutions as you go. Regular review and active practice are key to retention.

Q: Is this Linear Equations and Inequalities study guide free?

Yes — all study notes, flashcards, and practice problems for Linear Equations and Inequalities on Study Mondo are free to access. No account is needed.

Q: What course covers Linear Equations and Inequalities?

Linear Equations and Inequalities is part of the SAT Prep course on Study Mondo, specifically in the Algebra section. You can explore the full course for more related topics and practice resources.

Q: Are there practice problems for Linear Equations and Inequalities?

Yes, this page includes 16 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.

Solve linear equations and inequalities - core SAT skill

Written and reviewed by the Study Mondo Education TeamLast updated April 18, 2026

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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.

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: $y - y_1 = m(x - x_1)$

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

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: $x = 7$

Multi-Step Equations

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

Step-by-Step Strategy

Distribute any parentheses
Combine like terms on each side
Move variables to one side
Move constants to the other side
Divide to solve

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

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: $4x + 3x = 84$ $7x = 84$ $x = 12$

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

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: $-1 < x \leq 3$

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.

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)	$<$

Common SAT Mistakes

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

Quick Reference: 5-Step SAT Strategy

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

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)

📚 Practice Problems

1Problem 1easy

❓ Question:

If $4x - 7 = 21$ , what is the value of $x$ ?

💡 Show Solution

Solution:

$4x - 7 = 21$

Add 7 to both sides: $4x = 28$

Divide by 4: $x = 7$

Answer: $x = 7$

SAT Tip: Always check: $4(7) - 7 = 28 - 7 = 21$ ✓

2Problem 2easy

❓ Question:

If $4x - 7 = 21$ , what is the value of $x$ ?

💡 Show Solution

Solution:

$4x - 7 = 21$

Add 7 to both sides: $4x = 28$

Divide by 4: $x = 7$

Answer: $x = 7$

SAT Tip: Always check: $4(7) - 7 = 28 - 7 = 21$ ✓

3Problem 3medium

❓ Question:

Solve the system: $\begin{cases} y = 3x - 2 \\ 2x + y = 13 \end{cases}$

💡 Show Solution

Solution:

Use substitution - plug first equation into second: $2x + (3x - 2) = 13$ $5x - 2 = 13$ $5x = 15$ $x = 3$

Find $y$ : $y = 3(3) - 2 = 7$

Answer: $(3, 7)$ or $x = 3, y = 7$

Check: $2(3) + 7 = 13$ ✓

4Problem 4medium

❓ Question:

Solve the system: $\begin{cases} y = 3x - 2 \\ 2x + y = 13 \end{cases}$

💡 Show Solution

Solution:

Use substitution - plug first equation into second: $2x + (3x - 2) = 13$ $5x - 2 = 13$ $5x = 15$ $x = 3$

Find $y$ : $y = 3(3) - 2 = 7$

Answer: $(3, 7)$ or $x = 3, y = 7$

Check: $2(3) + 7 = 13$ ✓

5Problem 5hard

❓ Question:

A phone plan costs $\$ 25 $per month plus$ $0.10 $per text message. If the total bill was$ $37$, how many text messages were sent?

💡 Show Solution

Solution:

Let $t$ = number of text messages

Equation: $25 + 0.10t = 37$

Subtract 25: $0.10t = 12$

Divide by 0.10: $t = 120$

Answer: 120 text messages

SAT Tip: Set up word problems carefully - identify what the variable represents!

6Problem 6hard

❓ Question:

A phone plan costs $\$ 25 $per month plus$ $0.10 $per text message. If the total bill was$ $37$, how many text messages were sent?

💡 Show Solution

Solution:

Let $t$ = number of text messages

Equation: $25 + 0.10t = 37$

Subtract 25: $0.10t = 12$

Divide by 0.10: $t = 120$

Answer: 120 text messages

SAT Tip: Set up word problems carefully - identify what the variable represents!

7Problem 7easy

❓ Question:

If $5x - 3 = 22$ , what is the value of $x$ ?

💡 Show Solution

Step 1: Add 3 to both sides $5x - 3 + 3 = 22 + 3$ $5x = 25$

Step 2: Divide both sides by 5 $x = \frac{25}{5} = 5$

Check: $5(5) - 3 = 25 - 3 = 22$ ✓

Answer: $x = 5$

8Problem 8easy

❓ Question:

If $5x - 3 = 22$ , what is the value of $x$ ?

💡 Show Solution

Step 1: Add 3 to both sides $5x - 3 + 3 = 22 + 3$ $5x = 25$

Step 2: Divide both sides by 5 $x = \frac{25}{5} = 5$

Check: $5(5) - 3 = 25 - 3 = 22$ ✓

Answer: $x = 5$

9Problem 9easy

❓ Question:

If $-2x \geq 14$ , what is the greatest possible integer value of $x$ ?

💡 Show Solution

Step 1: Divide both sides by $-2$ and flip the inequality $x \leq \frac{14}{-2}$ $x \leq -7$

Key: We flip the inequality because we divided by a negative number.

The greatest integer less than or equal to $-7$ is $-7$ .

Answer: $x = -7$

10Problem 10easy

❓ Question:

If $-2x \geq 14$ , what is the greatest possible integer value of $x$ ?

💡 Show Solution

Step 1: Divide both sides by $-2$ and flip the inequality $x \leq \frac{14}{-2}$ $x \leq -7$

Key: We flip the inequality because we divided by a negative number.

The greatest integer less than or equal to $-7$ is $-7$ .

Answer: $x = -7$

11Problem 11medium

❓ Question:

If $3(2x - 4) + 7 = 4x + 5$ , what is the value of $x$ ?

💡 Show Solution

Step 1: Distribute the 3 $6x - 12 + 7 = 4x + 5$

Step 2: Combine like terms on the left $6x - 5 = 4x + 5$

Step 3: Subtract $4x$ from both sides $2x - 5 = 5$

Step 4: Add 5 to both sides $2x = 10$

Step 5: Divide by 2 $x = 5$

Check: $3(2(5) - 4) + 7 = 3(6) + 7 = 25$ and $4(5) + 5 = 25$ ✓

Answer: $x = 5$

12Problem 12medium

❓ Question:

If $3(2x - 4) + 7 = 4x + 5$ , what is the value of $x$ ?

💡 Show Solution

Step 1: Distribute the 3 $6x - 12 + 7 = 4x + 5$

Step 2: Combine like terms on the left $6x - 5 = 4x + 5$

Step 3: Subtract $4x$ from both sides $2x - 5 = 5$

Step 4: Add 5 to both sides $2x = 10$

Step 5: Divide by 2 $x = 5$

Check: $3(2(5) - 4) + 7 = 3(6) + 7 = 25$ and $4(5) + 5 = 25$ ✓

Answer: $x = 5$

13Problem 13hard

❓ Question:

If $\frac{2x + 3}{4} - \frac{x - 1}{3} = 2$ , what is the value of $10x - 5$ ?

💡 Show Solution

Step 1: Find the LCD of 4 and 3, which is 12. Multiply every term by 12. $12 \cdot \frac{2x + 3}{4} - 12 \cdot \frac{x - 1}{3} = 12 \cdot 2$ $3(2x + 3) - 4(x - 1) = 24$

Step 2: Distribute $6x + 9 - 4x + 4 = 24$

Step 3: Combine like terms $2x + 13 = 24$

Step 4: Solve for $x$ $2x = 11$ $x = \frac{11}{2}$

Step 5: Find what the question asks: $10x - 5$ $10\left(\frac{11}{2}\right) - 5 = 55 - 5 = 50$

Answer: $10x - 5 = 50$

SAT Tip: The question asks for $10x - 5$ , not $x$ . Always read carefully!

14Problem 14hard

❓ Question:

If $\frac{2x + 3}{4} - \frac{x - 1}{3} = 2$ , what is the value of $10x - 5$ ?

💡 Show Solution

Step 1: Find the LCD of 4 and 3, which is 12. Multiply every term by 12. $12 \cdot \frac{2x + 3}{4} - 12 \cdot \frac{x - 1}{3} = 12 \cdot 2$ $3(2x + 3) - 4(x - 1) = 24$

Step 2: Distribute $6x + 9 - 4x + 4 = 24$

Step 3: Combine like terms $2x + 13 = 24$

Step 4: Solve for $x$ $2x = 11$ $x = \frac{11}{2}$

Step 5: Find what the question asks: $10x - 5$ $10\left(\frac{11}{2}\right) - 5 = 55 - 5 = 50$

Answer: $10x - 5 = 50$

SAT Tip: The question asks for $10x - 5$ , not $x$ . Always read carefully!

15Problem 15expert

❓ Question:

For what value of $a$ does the equation $a(x + 2) - 3 = 4x + 2a - 7$ have no solution?

💡 Show Solution

Step 1: Distribute $a$ on the left $ax + 2a - 3 = 4x + 2a - 7$

Step 2: Subtract $2a$ from both sides $ax - 3 = 4x - 7$

Step 3: For NO solution, the $x$ coefficients must be equal but the constants must differ. Set the $x$ coefficients equal: $a = 4$

Step 4: Verify with $a = 4$ : $4x - 3 = 4x - 7$ $-3 = -7 \quad \text{(FALSE — no solution ✓)}$

Answer: $a = 4$

Why this works: When $a = 4$ , both sides have $4x$ , but the constants ( $-3$ vs $-7$ ) don't match, making the equation impossible.

16Problem 16expert

❓ Question:

For what value of $a$ does the equation $a(x + 2) - 3 = 4x + 2a - 7$ have no solution?

💡 Show Solution

Step 1: Distribute $a$ on the left $ax + 2a - 3 = 4x + 2a - 7$

Step 2: Subtract $2a$ from both sides $ax - 3 = 4x - 7$

Step 3: For NO solution, the $x$ coefficients must be equal but the constants must differ. Set the $x$ coefficients equal: $a = 4$

Step 4: Verify with $a = 4$ : $4x - 3 = 4x - 7$ $-3 = -7 \quad \text{(FALSE — no solution ✓)}$

Answer: $a = 4$

Why this works: When $a = 4$ , both sides have $4x$ , but the constants ( $-3$ vs $-7$ ) don't match, making the equation impossible.

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❓ Frequently Asked Questions

What is Linear Equations and Inequalities?▾

Solve linear equations and inequalities - core SAT skill

How can I study Linear Equations and Inequalities effectively?▾

Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 16 problems provided, checking solutions as you go. Regular review and active practice are key to retention.

Is this Linear Equations and Inequalities study guide free?▾

Yes — all study notes, flashcards, and practice problems for Linear Equations and Inequalities on Study Mondo are free to access. No account is needed.

What course covers Linear Equations and Inequalities?▾

Linear Equations and Inequalities is part of the SAT Prep course on Study Mondo, specifically in the Algebra section. You can explore the full course for more related topics and practice resources.

Are there practice problems for Linear Equations and Inequalities?▾

Yes, this page includes 16 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.

Linear Equations and Inequalities

Try the Interactive Version!

Linear Equations and Inequalities on the SAT

Why This Topic Matters

What Is a Linear Equation?

Solving One-Step Equations

Solving Two-Step Equations

Multi-Step Equations

Step-by-Step Strategy

Clearing Fractions

Special Cases

No Solution

Infinitely Many Solutions

Linear Inequalities

The One Critical Rule

Compound Inequalities

SAT-Specific Question Types

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

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

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

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

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

Word Problem Translation

Common SAT Mistakes

Quick Reference: 5-Step SAT Strategy

Practice Tips

📚 Practice Problems

1Problem 1easy

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2Problem 2easy

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3Problem 3medium

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4Problem 4medium

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5Problem 5hard

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6Problem 6hard

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7Problem 7easy

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8Problem 8easy

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9Problem 9easy

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10Problem 10easy

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11Problem 11medium

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12Problem 12medium

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13Problem 13hard

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14Problem 14hard

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15Problem 15expert

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16Problem 16expert

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❓ Frequently Asked Questions

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

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