Easy SAT Inequalities Practice Questions

Mastering Easy SAT Inequalities Practice Questions is a fundamental step toward achieving a high score on the math section of the SAT. Inequalities appear frequently in both the calculator and no-calculator portions, testing your ability to represent relationships between quantities that are not necessarily equal. By practicing these foundational problems, you build the speed and accuracy needed for more complex Easy SAT Algebra Practice Questions that you will encounter on test day.

Concept Explanation

SAT inequalities are mathematical statements that use symbols like $<$, $>$, $\leq$, or $\geq$ to compare two expressions instead of an equals sign. To solve a linear inequality, you follow the same inverse operation rules used in basic algebra, such as adding or subtracting terms from both sides to isolate the variable. The most critical rule to remember is that whenever you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. For example, if $-2x < 10$, dividing by $-2$ results in $x > -5$. On the SAT, you will often be asked to solve for a variable, identify a possible value in a solution set, or translate a real-world word problem into an inequality. Understanding how to graph these solutions on a number line—where open circles represent $<$ or $>$ and closed circles represent $\leq$ or $\geq$—is also essential for visual questions. For more foundational practice, check out our guide on Easy SAT Math Practice Questions.

Solved Examples

1. Solve for $x$: $$3x - 5 > 10$$
   1. Add 5 to both sides: $3x > 15$.
   2. Divide both sides by 3: $x > 5$.
   3. The solution is all values of $x$ greater than 5.
2. Solve for $y$: $$-4y + 2 \leq 14$$
   1. Subtract 2 from both sides: $-4y \leq 12$.
   2. Divide both sides by $-4$. Since we are dividing by a negative number, flip the inequality sign: $y \geq -3$.
   3. The solution is all values of $y$ greater than or equal to $-3$.
3. Word Problem: A taxi service charges a flat fee of $3 plus $2 per mile. If Sarah wants to spend no more than $15 on her ride, what is the maximum number of miles $m$ she can travel?
   1. Set up the inequality: $3 + 2m \leq 15$.
   2. Subtract 3 from both sides: $2m \leq 12$.
   3. Divide by 2: $m \leq 6$.
   4. The maximum number of miles is 6.

Practice Questions

1. Solve the inequality for $x$: $$5x + 7 < 22$$

2. Which of the following values is a solution to the inequality $$\frac{x}{3} - 4 \geq -2$$?
A) 3
B) 4
C) 5
D) 6

3. Solve for $w$: $$12 - 2w > 4$$

4. A movie theater charges $10 per ticket. A group has a coupon for $5 off their total purchase. If they want to spend at most $45, what is the maximum number of tickets $t$ they can buy?

5. Solve the inequality: $$7 - x \leq 10$$

6. If $2x + 5 > 11$, what is the smallest integer value $x$ could be?

7. Solve for $z$: $$\frac{3}{4}z + 1 \leq 10$$

8. Which inequality represents "the sum of 4 and twice a number $n$ is at least 18"?
A) $4 + 2n < 18$
B) $4 + 2n \leq 18$
C) $4 + 2n > 18$
D) $4 + 2n \geq 18$

9. Solve for $k$: $$-5k - 3 < 12$$

10. A worker earns $15 per hour. They need to earn at least $120 today to cover expenses. If they have already earned $30, how many more hours $h$ must they work?

Answers & Explanations

1. Answer: $x < 3$. Subtract 7 from both sides to get $5x < 15$. Divide by 5 to get $x < 3$.
2. Answer: D. Add 4 to both sides: $\frac{x}{3} \geq 2$. Multiply by 3: $x \geq 6$. Only option D (6) satisfies this.
3. Answer: $w < 4$. Subtract 12 from both sides: $-2w > -8$. Divide by $-2$ and flip the sign: $w < 4$.
4. Answer: 5. The inequality is $10t - 5 \leq 45$. Add 5: $10t \leq 50$. Divide by 10: $t \leq 5$.
5. Answer: $x \geq -3$. Subtract 7 from both sides: $-x \leq 3$. Multiply by $-1$ and flip the sign: $x \geq -3$.
6. Answer: 4. Subtract 5: $2x > 6$. Divide by 2: $x > 3$. The smallest integer greater than 3 is 4.
7. Answer: $z \leq 12$. Subtract 1: $\frac{3}{4}z \leq 9$. Multiply by $\frac{4}{3}$: $z \leq 12$.
8. Answer: D. "The sum of 4 and twice a number" is $4 + 2n$. "At least" means greater than or equal to ($\geq$).
9. Answer: $k > -3$. Add 3: $-5k < 15$. Divide by $-5$ and flip the sign: $k > -3$.
10. Answer: $h \geq 6$. The inequality is $30 + 15h \geq 120$. Subtract 30: $15h \geq 90$. Divide by 15: $h \geq 6$.

Quick Quiz

Frequently Asked Questions

What is the golden rule of inequalities?

The golden rule of inequalities states that you must flip the inequality sign whenever you multiply or divide both sides of the expression by a negative number. This ensures the mathematical relationship remains true on the number line.

Does the SAT focus more on linear or quadratic inequalities?

The SAT primarily focuses on linear inequalities, especially in the "Heart of Algebra" section. While quadratic inequalities do appear, they are less common than linear systems and basic one-variable inequalities.

How do I graph an inequality with a 'greater than or equal to' sign?

To graph $\geq$, you place a solid (closed) circle on the starting number to show it is included in the solution. Then, you shade the number line to the right to represent all values larger than that number.

What does 'no more than' mean in SAT word problems?

In SAT word problems, the phrase 'no more than' translates to the less than or equal to symbol ($\leq$). It indicates a maximum limit that the quantity can reach but not exceed.

Can I use a calculator for all inequality questions?

No, some inequality questions appear in the Section 3 (No-Calculator) portion of the SAT. It is important to practice solving them manually to ensure you can handle basic arithmetic and sign flips quickly. For more practice without a calculator, see our SAT Algebra Practice Questions with Answers.

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