# SAT Inequalities Worksheet PDF with Answers | 1600.now Source: https://1600.now/sat-linear-inequalities-worksheet --- ## SAT Inequalities Worksheet Solve a linear inequality with the same algebra as an equation, except that multiplying or dividing by a negative number reverses the inequality sign. Then check the endpoint and the direction of the solution set. Written by [Luke Finigan](https://1600.now/about) This 12-question worksheet covers strict and inclusive limits, budgets, whole-number constraints, and systems. Print a student copy or download the PDF with explained answers. 12 questions · about 25 minutes · Algebra › Linear inequalities in one or two variables [Download 4-page PDF](https://1600.now/downloads/worksheets/sat-linear-inequalities-worksheet-student-letter.pdf) [Share to Google Classroom](https://classroom.google.com/share?url=https%3A%2F%2F1600.now%2Fsat-linear-inequalities-worksheet) · This worksheet is free to print and share under [CC BY-NC-ND 4.0](https://creativecommons.org/licenses/by-nc-nd/4.0/). Tutors may also use it in paid sessions. If you post it online, please link to this page instead of re-uploading the PDF. These are original questions, not College Board items. ### Test the boundary and an allowed value An addition or subtraction leaves the inequality direction unchanged. Multiplying or dividing by a negative reverses it: $-2x\ge10$ gives $x\le-5$. Test $x=-6$ and $x=0$ in the original to confirm which side works. A strict sign, $<$ or $>$, excludes the endpoint; $\le$ or $\ge$ includes it. A point on the line $y=2x+1$ does not satisfy $y>2x+1$, because equality is excluded. For whole-number amounts, use the question’s limit to choose how to round. A maximum count cannot exceed a budget; a minimum number of work hours must meet the earnings target. Substitute the chosen integer and its neighboring integer to check the boundary. A point in a system must satisfy every inequality. Test each condition separately instead of stopping after the first one passes. Try it yourself: Which inequality is equivalent to $-3x<12$? · A $x<-4$ B $x>-4$ C $x>4$ ### Questions 1. **Question 1** Which inequality is equivalent to $3x + 5 < 20$? A. $x < 5$ B. $x > 5$ C. $x < 15$ D. $x > 15$ **A.** Subtract 5 and divide by positive 3: $3x < 15$ becomes $x < 5$. The inequality direction stays the same. 2. **Question 2** Which inequality is equivalent to $-2x \ge 10$? A. $x \ge -5$ B. $x \le -5$ C. $x \ge 5$ D. $x \le 5$ **B.** Divide by $-2$ and reverse the inequality direction. The result is $x \le -5$. 3. **Question 3** A club has $120. It spends $30 on supplies and $9 per ticket. What is the maximum number of tickets it can buy? A. 9 B. 10 C. 13 D. 16 **B.** The budget inequality is $30+9t\le 120$. Thus $t\le 10$, and ten tickets can be purchased. 4. **Question 4** Which value satisfies both $x>-2$ and $x\le 3$? A. −3 B. −2 C. 2 D. 4 **C.** The interval includes numbers greater than −2 and at most 3. Only 2 meets both conditions. 5. **Question 5** Which point satisfies $y>2x+1$? A. (0, 1) B. (1, 2) C. (2, 6) D. (3, 7) **C.** At $x=2$, the boundary has $y=5$, and $6>5$. The other points are on or below the boundary. 6. **Question 6** If $5-x\le 8$, which inequality describes $x$? A. $x\le -3$ B. $x\ge -3$ C. $x\le 3$ D. $x\ge 3$ **B.** Subtract 5 to get $-x\le 3$. Dividing by −1 reverses the sign, giving $x\ge -3$. 7. **Question 7** A truck can carry at most 900 kilograms. A machine weighs 260 kilograms and each box weighs 40 kilograms. What is the maximum number of boxes it can also carry? A. 15 B. 16 C. 17 D. 22 **B.** There are $900-260=640$ kilograms available. Dividing by 40 gives 16 boxes. 8. **Question 8** Which inequality is equivalent to $2(x-4)>x+3$? A. $x>-11$ B. $x<-11$ C. $x>11$ D. $x<11$ **C.** Expand to $2x-8>x+3$. Subtract $x$ and add 8 to obtain $x>11$. 9. **Question 9** An integer $n$ satisfies $4n+1<18$. What is the largest possible value of $n$? A. 3 B. 4 C. 5 D. 17 **B.** The inequality gives $n<\dfrac{17}{4}=4.25$. The largest integer below 4.25 is 4. 10. **Question 10** Which point satisfies both $x+y\le 6$ and $y\ge 2$? A. (5, 2) B. (2, 1) C. (3, 3) D. (4, 4) **C.** For (3, 3), $x+y=6$ and $y=3\ge 2$. Each other point violates at least one condition. 11. **Question 11** A student earns $12 per hour. What is the least number of whole hours the student must work to earn at least $100? A. 7 B. 8 C. 9 D. 10 **C.** The inequality $12h\ge 100$ gives $h\ge 8\dfrac{1}{3}$. Nine is the smallest whole number satisfying the requirement. 12. **Question 12** Which inequality is equivalent to $\dfrac{x+6}{-3}<2$? A. $x<-12$ B. $x>-12$ C. $x<0$ D. $x>0$ **B.** Multiplying by −3 reverses the inequality: $x+6>-6$. Subtracting 6 gives $x>-12$. 0/12 correct ### Check your inequalities answers Each row is a wrong answer choice from one of the sheet's problems. Find the one you picked, then run the check in the last column. | What your answer looked like | What actually happened | Fix it next time | | --- | --- | --- | | $x \ge -5$ for $-2x \ge 10$ (problem 2) | You divided by $-2$ without reversing the inequality. | Dividing by a negative flips the sign: $x \le -5$. Test $x = 0$, which the wrong answer allows: $-2(0) = 0$, and 0 isn't at least 10. | | 13 in problem 3 or 22 in problem 7 | You divided the whole limit by the cost or weight of one item and skipped the fixed part: the $30 of supplies, or the 260-kilogram machine. | Subtract the fixed amount first. Problem 3: $\dfrac{120 - 30}{9} = 10$ tickets. Problem 7: $\dfrac{900 - 260}{40} = 16$ boxes. | | $-2$ in problem 4 | $x > -2$ is strict, so $-2$ itself is left out. | Read each endpoint: $>$ and $<$ leave it out, and $\ge$ and $\le$ include it. Of the choices, only 2 is greater than $-2$ and at most 3. | | $(3, 7)$ or $(0, 1)$ in problem 5 | Both points are on the line $y = 2x + 1$: $2(3) + 1 = 7$ and $2(0) + 1 = 1$. The inequality $y > 2x + 1$ needs points strictly above it. | Substitute and compare. For $(2, 6)$, $2(2) + 1 = 5$, and $6 > 5$. | | 5 in problem 9 or 8 in problem 11 | You rounded the wrong way. In problem 9, $n < 4.25$, and $n = 5$ breaks it: $4(5) + 1 = 21$. In problem 11, 8 hours earns $96, which is less than $100. | Put your whole number back into the original inequality. The answers are 4 in problem 9 and 9 in problem 11. | | $(5, 2)$ in problem 10 | $(5, 2)$ meets $y \ge 2$ but not $x + y \le 6$, since $5 + 2 = 7$. | A point has to satisfy every inequality in the system. Check both for each choice. Only $(3, 3)$ passes: $3 + 3 = 6$ and $3 \ge 2$. | #### Worked example: dividing by a negative Problem 12 asks which inequality is equivalent to $\dfrac{x + 6}{-3} < 2$. Multiply both sides by $-3$ to clear the fraction. Multiplying by a negative reverses the inequality, so $<$ becomes $>$: $x + 6 > -6$. Subtract 6 to get $x > -12$. Test one value on each side of $-12$. At $x = 0$, $\dfrac{6}{-3} = -2$, and $-2 < 2$ is true. At $x = -15$, $\dfrac{-9}{-3} = 3$, and $3 < 2$ is false. So the solutions are the numbers greater than $-12$. The choice $x < -12$ keeps the original sign and includes $-15$. ### Apply the check to fresh questions For another inequality, test an endpoint and a value on each side. For a count, check the next integer too. Continue with the inequalities practice link. [Practice inequalities questions](https://1600.now/bank/math/skill/Linear%20inequalities%20in%20one%20or%20two%20variables) [Review the linear inequalities method](https://1600.now/sat-skill/linear-inequalities) ### More Math worksheets [All worksheets](https://1600.now/sat-worksheets) - [Linear Equations](https://1600.now/sat-linear-equations-one-variable-worksheet) - [Linear Equations in Two Variables](https://1600.now/sat-linear-equations-two-variables-worksheet) - [Systems of Equations](https://1600.now/sat-systems-of-linear-equations-worksheet)