Solve Linear Equations - HSPT Math

Solve for $a$ : $7(a-2)+3=2(5a-1).$
Answer: $a=-3$ When you encounter a linear equation with variables on both sides, your goal is to isolate the variable by using the distributive property and combining like terms systematically. Start by distributing on both sides: $7(a-2)+3=2(5a-1)$ becomes $7a-14+3=10a-2$ . Simplify the left side: $7a-11=10a-2$ . Next, collect all terms with $a$ on one side and constants on the other. Subtract $7a$ from both sides: $-11=3a-2$ . Add $2$ to both sides: $-9=3a$ . Finally, divide by $3$ : $a=-3$ . Let's verify: $7(-3-2)+3=7(-5)+3=-35+3=-32$ and $2(5(-3)-1)=2(-15-1)=2(-16)=-32$ . Both sides equal $-32$ , confirming our answer. Looking at the wrong choices: Choice B ( $a=3$ ) likely comes from sign errors during distribution or when moving terms across the equation. Choice C ( $a=-\frac{9}{2}$ ) results from forgetting to divide by $3$ in the final step, stopping at $-9=3a$ . Choice D ( $a=\frac{9}{2}$ ) combines both errors—incorrect signs and incomplete solving. Study tip: Always verify your solution by substituting back into the original equation. This catches algebraic errors and ensures you haven't fallen into common traps like sign mistakes or incomplete operations. Work methodically through each step, and remember that both sides of the equation should yield the same value when you substitute your answer.
If $\frac{x}{3} + \frac{x}{4} = \frac{x}{2} - 5$ , what is the value of $x$ ?
Answer: $x = -60$ When you encounter an equation with fractions containing the same variable, your goal is to combine like terms and isolate the variable. Start by finding a common denominator to simplify the fractions on the left side. To solve $\frac{x}{3} + \frac{x}{4} = \frac{x}{2} - 5$ , first combine the left side using the common denominator 12: $\frac{4x}{12} + \frac{3x}{12} = \frac{7x}{12}$ Now your equation becomes: $\frac{7x}{12} = \frac{x}{2} - 5$ Convert $\frac{x}{2}$ to twelfths: $\frac{x}{2} = \frac{6x}{12}$ So: $\frac{7x}{12} = \frac{6x}{12} - 5$ Subtract $\frac{6x}{12}$ from both sides: $\frac{x}{12} = -5$ Multiply both sides by 12: $x = -60$ Choice B ( $x = -60$ ) is correct. You can verify by substituting back: $\frac{-60}{3} + \frac{-60}{4} = -20 + (-15) = -35$ , and $\frac{-60}{2} - 5 = -30 - 5 = -35$ ✓ Choice A ( $x = -50$ ) gives you $-\frac{65}{6}$ on the left and $-30$ on the right—not equal. Choice C ( $x = -70$ ) produces $-\frac{175}{12}$ versus $-40$ —also unequal. Choice D ( $x = -80$ ) yields $-\frac{200}{12}$ compared to $-45$ —incorrect as well. When solving fraction equations, always find common denominators first, then systematically eliminate fractions by multiplying through. Double-check your answer by substituting back into the original equation—this catches arithmetic errors that are easy to make with fraction calculations.
Solve for $x$ : $3(2x-5)+4 = 2(x+7)-2.$
Answer: $x=\dfrac{23}{4}$ This equation tests your ability to solve linear equations with parentheses and variables on both sides. When you encounter equations like this, your goal is to systematically eliminate parentheses, combine like terms, and isolate the variable. Start by distributing on both sides: $3(2x-5)+4 = 2(x+7)-2$ becomes $6x-15+4 = 2x+14-2$ . Simplifying the constants gives you $6x-11 = 2x+12$ . Next, collect all terms with $x$ on one side and constants on the other. Subtract $2x$ from both sides: $4x-11 = 12$ . Add 11 to both sides: $4x = 23$ . Finally, divide by 4: $x = \frac{23}{4}$ . Choice A, $x=\frac{23}{4}$ , is correct. Choice B, $x=\frac{21}{4}$ , likely results from an arithmetic error when combining constants—perhaps getting $4x = 21$ instead of $4x = 23$ . Choice C, $x=\frac{25}{4}$ , could come from incorrectly handling the signs when distributing, leading to $4x = 25$ . Choice D, $x=\frac{23}{5}$ , suggests an error in collecting like terms, possibly getting $5x = 23$ instead of $4x = 23$ . Remember to work methodically: distribute first, combine like terms on each side, then move all variable terms to one side and constants to the other. Double-check your arithmetic at each step, especially when dealing with negative numbers and distribution. These multi-step equations reward careful, systematic work over speed.
Maria has 3 times as many books as Carlos. If Maria gives Carlos 12 books, they will have equal numbers of books. How many books does Maria have originally?
Answer: Maria originally has 36 books When you encounter word problems involving relationships between quantities that change, setting up equations with variables is your most reliable approach. These problems test your ability to translate verbal descriptions into mathematical expressions. Let's define variables: Let $C$ = Carlos's original number of books and $M$ = Maria's original number of books. From the problem, we know two key relationships. First, "Maria has 3 times as many books as Carlos," which gives us $M = 3C$ . Second, "If Maria gives Carlos 12 books, they will have equal numbers," meaning Maria will have $M - 12$ books and Carlos will have $C + 12$ books, so $M - 12 = C + 12$ . From the second equation: $M - 12 = C + 12$ , so $M = C + 24$ . Now substitute the first equation: $3C = C + 24$ . Solving: $2C = 24$ , so $C = 12$ . Therefore, $M = 3(12) = 36$ . Let's verify: Maria starts with 36, Carlos with 12. After the transfer, Maria has 24 and Carlos has 24. ✓ Choice A is correct with 36 books. Choice B (42 books) would mean Carlos originally had 14 books, but after the transfer they'd have 30 and 26 respectively—not equal. Choice C (48 books) implies Carlos had 16 originally, leading to 36 and 28 after transfer. Choice D (54 books) means Carlos had 18, resulting in 42 and 30 after transfer. Always verify your solution by checking it against both original conditions. This catches algebraic errors and confirms your setup was correct.
Solve for $y$ : $4(2y-1)-3y = 5.$
Answer: $y=\dfrac{9}{5}$ When you encounter a linear equation with parentheses and multiple terms, your goal is to systematically simplify by following the order of operations, then isolate the variable. Start by distributing the 4 to both terms inside the parentheses: $4(2y-1) = 4 \cdot 2y - 4 \cdot 1 = 8y - 4$ . The equation becomes: $8y - 4 - 3y = 5$ . Next, combine like terms on the left side: $8y - 3y = 5y$ , so you have $5y - 4 = 5$ . Add 4 to both sides: $5y = 9$ . Finally, divide both sides by 5: $y = \frac{9}{5}$ . This confirms answer choice A is correct. Let's examine why the other answers are wrong. Answer B ( $y = \frac{7}{5}$ ) likely results from incorrectly combining terms, perhaps getting $3y - 4 = 5$ instead of $5y - 4 = 5$ . Answer C ( $y = \frac{11}{5}$ ) might come from an error in distributing, like getting $6y - 4 = 5$ and solving for $y = \frac{9}{6} = \frac{11}{5}$ through incorrect arithmetic. Answer D ( $y = 2$ ) could result from multiple errors, such as incorrectly distributing and then making computational mistakes. Always verify your answer by substituting back into the original equation. With $y = \frac{9}{5}$ : $4(2 \cdot \frac{9}{5} - 1) - 3 \cdot \frac{9}{5} = 4(\frac{18}{5} - 1) - \frac{27}{5} = 4 \cdot \frac{13}{5} - \frac{27}{5} = \frac{52}{5} - \frac{27}{5} = \frac{25}{5} = 5$ ✓
Maria already has 12 stamps and purchases 3 more each week. The equation $12+3w=45$ shows when she will own 45 stamps. How many weeks, $w$ , will that take?
Answer: $11$ This is a linear equation problem where you need to solve for the variable $w$ . When you see an equation already set up like this, your job is to isolate the variable on one side. Starting with $12 + 3w = 45$ , you need to get $w$ by itself. First, subtract 12 from both sides to eliminate the constant term on the left: $3w = 45 - 12 = 33$ . Then divide both sides by 3 to isolate $w$ : $w = 33 ÷ 3 = 11$ . So Maria will have 45 stamps after 11 weeks. Let's check why the other answers are wrong. Choice B (9 weeks) would give Maria $12 + 3(9) = 12 + 27 = 39$ stamps—not enough. Choice C (13 weeks) would result in $12 + 3(13) = 12 + 39 = 51$ stamps—too many. Choice D (7 weeks) would only give her $12 + 3(7) = 12 + 21 = 33$ stamps—far short of her goal. Choice A (11 weeks) is correct because $12 + 3(11) = 12 + 33 = 45$ stamps exactly. Remember to always verify your answer by substituting it back into the original equation. This catches arithmetic errors and confirms you solved correctly. On linear equation problems like this, the HSPT often includes answer choices that result from common mistakes like forgetting to subtract the initial amount or making simple calculation errors.
Solve the inequality $3(x - 4) + 2(x + 1) \leq 5x - 7$ and express the solution in interval notation.
Answer: All real numbers When solving linear inequalities, you follow the same steps as solving equations, but you must be careful about when to flip the inequality sign. Let's work through this step-by-step. First, expand both sides: $3(x - 4) + 2(x + 1) = 3x - 12 + 2x + 2 = 5x - 10$ . So the inequality becomes $5x - 10 \leq 5x - 7$ . Next, subtract $5x$ from both sides: $5x - 10 - 5x \leq 5x - 7 - 5x$ , which simplifies to $-10 \leq -7$ . Here's the key insight: we now have a statement with no variables. Since $-10$ is indeed less than $-7$ , this statement is always true, regardless of what value $x$ takes. This means every real number satisfies the original inequality. Looking at the wrong answers: Choice A) $(-\infty, 3]$ suggests the solution stops at $x = 3$ , but there's nothing special about this value in our work. Choice B) $(-\infty, 5]$ similarly restricts the solution unnecessarily to $x \leq 5$ . Choice C) $[3, \infty)$ not only picks an arbitrary starting point but also excludes negative numbers, which we know satisfy the inequality. The correct answer is D) All real numbers, since our simplified inequality $-10 \leq -7$ is universally true. Study tip: When solving inequalities, if you eliminate all variables and get a true statement (like $-10 \leq -7$ ), the solution is all real numbers. If you get a false statement, there's no solution.
Find $n$ : $3(2n-1)-(n+4)=2(n-3).$
Answer: $n=\dfrac13$ This is a linear equation that requires careful distribution and combining of like terms. When you encounter equations with parentheses on both sides, your goal is to systematically simplify each side, then isolate the variable. Start by distributing on the left side: $3(2n-1) = 6n - 3$ and $-(n+4) = -n - 4$ , so the left side becomes $6n - 3 - n - 4 = 5n - 7$ . On the right side: $2(n-3) = 2n - 6$ . Your equation is now $5n - 7 = 2n - 6$ . Subtract $2n$ from both sides: $3n - 7 = -6$ . Add 7 to both sides: $3n = 1$ . Therefore, $n = \frac{1}{3}$ , which is choice A. Let's check why the other answers are wrong. Choice B ( $n = -\frac{1}{3}$ ) likely comes from a sign error when distributing or combining terms. Choice C ( $n = 1$ ) might result from incorrectly handling the fraction $\frac{1}{3}$ or making arithmetic errors in the final steps. Choice D ( $n = -1$ ) could come from multiple sign errors, particularly when distributing the negative sign in front of $(n+4)$ . You can verify A is correct by substituting back: $3(2 \cdot \frac{1}{3} - 1) - (\frac{1}{3} + 4) = 3(\frac{2}{3} - 1) - \frac{13}{3} = -1 - \frac{13}{3} = -\frac{16}{3}$ , and $2(\frac{1}{3} - 3) = 2(-\frac{8}{3}) = -\frac{16}{3}$ . ✓ Always double-check your distribution of negative signs and verify your answer by substitution—these are the most common sources of error in linear equations.
Solve the inequality $\dfrac{2x+1}{4}\le 3.$
Answer: $x\le \dfrac{11}{2}$ When you encounter an inequality with a fraction, your goal is to isolate the variable while preserving the inequality's direction. The key principle is that you can perform the same operations on both sides of an inequality as you would with an equation, as long as you don't multiply or divide by a negative number. Starting with $\frac{2x+1}{4} \le 3$ , multiply both sides by 4 to eliminate the fraction: $2x + 1 \le 12$ . Next, subtract 1 from both sides: $2x \le 11$ . Finally, divide both sides by 2: $x \le \frac{11}{2}$ . Since we only performed operations with positive numbers, the inequality direction stays the same. Let's examine why the other choices are incorrect. Choice B gives $x \ge \frac{11}{2}$ , which flips the inequality sign incorrectly—this would happen if you mistakenly thought dividing by positive 2 required flipping the sign. Choices C and D both show $x < -\frac{11}{2}$ and $x > -\frac{11}{2}$ respectively. These likely result from sign errors when subtracting 1, perhaps getting $2x \le -11$ instead of $2x \le 11$ . You can verify the correct answer A by testing a value: if $x = 5$ , then $\frac{2(5)+1}{4} = \frac{11}{4} = 2.75$ , and indeed $2.75 \le 3$ . Study tip: Remember that inequality signs only flip when you multiply or divide by a negative number. With fractions, focus on clearing denominators first by multiplying both sides by the denominator.
Solve for $x$ : $5-2(x-4)=3(x+1)+2.$
Answer: $x=\dfrac85$ When you encounter a linear equation with parentheses and variables on both sides, your goal is to systematically isolate the variable using the order of operations in reverse. Start by distributing through the parentheses. On the left side: $5-2(x-4) = 5-2x+8 = 13-2x$ . On the right side: $3(x+1)+2 = 3x+3+2 = 3x+5$ . Your equation becomes $13-2x = 3x+5$ . Next, collect all variable terms on one side and constants on the other. Add $2x$ to both sides: $13 = 5x+5$ . Subtract 5 from both sides: $8 = 5x$ . Finally, divide by 5: $x = \frac{8}{5}$ . Looking at the wrong answers: Choice B gives $x = \frac{13}{5}$ , which likely results from incorrectly handling the distribution or making sign errors when moving terms. Choice C gives $x = \frac{5}{8}$ , which happens when you accidentally flip the fraction at the end—dividing 5 by 8 instead of 8 by 5. Choice D gives $x = -\frac{8}{5}$ , which occurs from sign errors, particularly when distributing the negative through $-2(x-4)$ or when moving terms across the equals sign. The correct answer is A: $x = \frac{8}{5}$ . Study tip: When solving multi-step equations, work methodically through each step and double-check your distribution of negative signs. Always verify your answer by substituting it back into the original equation—both sides should equal the same value.
Solve for $x$ : $0.25(8x+12)-3 = 0.5x.$
Answer: $x=0$ When you encounter a linear equation with decimals and parentheses like this one, your goal is to systematically simplify both sides until you isolate the variable. The key is working carefully through the order of operations and decimal arithmetic. Start by distributing $0.25$ to both terms inside the parentheses: $0.25(8x + 12) = 0.25 \cdot 8x + 0.25 \cdot 12 = 2x + 3$ . Your equation becomes: $2x + 3 - 3 = 0.5x$ , which simplifies to $2x = 0.5x$ . Now subtract $0.5x$ from both sides: $2x - 0.5x = 0$ , giving you $1.5x = 0$ . When any number times $x$ equals zero, then $x = 0$ . Let's verify: substituting $x = 0$ into the original equation gives $0.25(0 + 12) - 3 = 3 - 3 = 0$ , and $0.5(0) = 0$ . Both sides equal zero, confirming our answer. Looking at the wrong choices: Choice B ( $x = 6$ ) likely comes from calculation errors in the distribution step. Choice C ( $x = -6$ ) might result from sign errors when moving terms between sides. Choice D ( $x = 3$ ) could come from confusing the simplified form $2x + 3 - 3$ and thinking the constant term somehow becomes the answer. The correct answer is A. Strategy tip: With decimal coefficients, convert them to fractions if needed ( $0.25 = \frac{1}{4}$ , $0.5 = \frac{1}{2}$ ) to avoid arithmetic mistakes, and always verify your solution by substituting back into the original equation.
Solve for $p$ : $\dfrac{4}{3}p-2=\dfrac{1}{6}(12p+6).$
Answer: $p=-\dfrac{9}{2}$ When you encounter an equation with fractions on both sides, your goal is to isolate the variable by simplifying and combining like terms systematically. Start by simplifying the right side. Distribute $\frac{1}{6}$ to both terms in the parentheses: $\frac{1}{6}(12p + 6) = \frac{12p}{6} + \frac{6}{6} = 2p + 1$ . Now your equation becomes: $\frac{4}{3}p - 2 = 2p + 1$ . Next, collect all terms with $p$ on one side and constants on the other. Subtract $2p$ from both sides: $\frac{4}{3}p - 2p - 2 = 1$ . To subtract $2p$ from $\frac{4}{3}p$ , convert $2p$ to thirds: $2p = \frac{6}{3}p$ . So $\frac{4}{3}p - \frac{6}{3}p = -\frac{2}{3}p$ . Your equation is now $-\frac{2}{3}p - 2 = 1$ . Add 2 to both sides: $-\frac{2}{3}p = 3$ . Finally, multiply both sides by $-\frac{3}{2}$ to solve for $p$ : $p = 3 \times (-\frac{3}{2}) = -\frac{9}{2}$ . This confirms answer A. Answer B ( $p = -\frac{9}{4}$ ) likely results from an error in fraction arithmetic when combining coefficients. Answer C ( $p = \frac{9}{2}$ ) comes from forgetting the negative sign when multiplying by the reciprocal. Answer D ( $p = -\frac{3}{2}$ ) suggests an error in distributing the fraction on the right side. Always double-check your work by substituting your answer back into the original equation to verify both sides are equal.
A chemist needs to create a 30% acid solution. How many liters of a 50% acid solution must be added to 10 liters of a 20% acid solution to obtain the desired mixture?
Answer: 5 liters Let $x$ be the number of liters of the 50% solution to be added. The total amount of acid from the two solutions must equal the amount of acid in the final mixture. Acid from the first solution: $0.20 \times 10 = 2$ liters. Acid from the second solution: $0.50 \times x$ . The total volume of the mixture will be $10 + x$ liters. The total acid in the final mixture will be $0.30(10 + x)$ . Set up the equation: $2 + 0.50x = 0.30(10 + x)$ . Distribute on the right side: $2 + 0.5x = 3 + 0.3x$ . Subtract $0.3x$ from both sides: $2 + 0.2x = 3$ . Subtract 2 from both sides: $0.2x = 1$ . Divide by 0.2: $x = 1 / 0.2 = 5$ . So, 5 liters must be added.
A number decreased by 15 is equal to 3 times the number increased by 9. What is the number?
Answer: The number is -12 When you see a word problem asking you to find an unknown number, you're dealing with algebraic translation. The key is converting the English phrases into mathematical expressions, then solving the resulting equation. Let's call the unknown number $x$ . "A number decreased by 15" translates to $x - 15$ . "3 times the number increased by 9" means $3x + 9$ . Since these expressions are equal, you get: $x - 15 = 3x + 9$ . To solve, collect like terms. Subtract $x$ from both sides: $-15 = 2x + 9$ . Then subtract 9 from both sides: $-24 = 2x$ . Finally, divide by 2: $x = -12$ . You can verify this: $-12 - 15 = -27$ and $3(-12) + 9 = -36 + 9 = -27$ ✓ Looking at the wrong answers: Choice A (-15) likely comes from focusing too much on the "decreased by 15" part without solving the full equation. Choice C (-9) might result from confusing the constants in the problem or making sign errors. Choice D (-6) could come from arithmetic mistakes when combining terms or dividing incorrectly. The correct answer is B. Remember this translation pattern: "decreased by" means subtraction, "increased by" means addition, and "times" indicates multiplication. Always check your answer by substituting back into the original word problem to ensure both sides truly equal each other.
Find the sum of all solutions for the equation $|2x + 3| = 4x$ .
Answer: 1.5 An absolute value equation $|A| = B$ requires solving two separate cases: $A = B$ and $A = -B$ . It also requires that $B \ge 0$ , so in this case $4x \ge 0$ , which means $x \ge 0$ . Any solution where $x$ is negative will be extraneous. Case 1: $2x + 3 = 4x$ . Subtracting $2x$ from both sides gives $3 = 2x$ , so $x = 1.5$ . This solution is valid since $1.5 \ge 0$ . Case 2: $2x + 3 = -4x$ . Subtracting $2x$ gives $3 = -6x$ , so $x = -0.5$ . This solution is extraneous because it does not satisfy the condition $x \ge 0$ . Therefore, there is only one valid solution, $x = 1.5$ . The sum of all solutions is just 1.5.
Solve for $k$ : $\dfrac{5-k}{3}=2-k.$
Answer: $k=\dfrac12$ This equation involves a fraction set equal to an expression, so you'll need to eliminate the fraction and solve systematically. The key is to clear the denominator first, then collect like terms. Start by multiplying both sides by 3 to eliminate the fraction: $3 \cdot \frac{5-k}{3} = 3(2-k)$ . This gives you $5-k = 6-3k$ . Now you have a standard linear equation without fractions. Next, collect the $k$ terms on one side. Add $3k$ to both sides: $5-k+3k = 6-3k+3k$ , which simplifies to $5+2k = 6$ . Subtract 5 from both sides: $2k = 1$ . Finally, divide by 2: $k = \frac{1}{2}$ . Let's examine why the other answers are incorrect. Choice B ( $k=2$ ) would result if you made an error when collecting terms, perhaps forgetting to distribute the negative sign properly. Choice C ( $k=-\frac{1}{2}$ ) comes from a sign error, likely when moving terms from one side to the other. Choice D ( $k=-2$ ) suggests multiple computational mistakes, possibly in both the fraction elimination and variable collection steps. You can verify: substituting $k=\frac{1}{2}$ gives $\frac{5-\frac{1}{2}}{3} = \frac{\frac{9}{2}}{3} = \frac{3}{2}$ , and $2-\frac{1}{2} = \frac{3}{2}$ ✓. When solving equations with fractions, always clear denominators first by multiplying both sides by the denominator. This prevents fraction arithmetic errors and makes the algebra much cleaner.
If $2|x - 3| + 5 = 17$ , what is the sum of all possible values of $x$ ?
Answer: 6 When you encounter absolute value equations, remember that the expression inside the absolute value bars can be either positive or negative, leading to multiple solutions. Start by isolating the absolute value term. From $2|x - 3| + 5 = 17$ , subtract 5 from both sides: $2|x - 3| = 12$ . Then divide by 2: $|x - 3| = 6$ . Now solve the two cases. Since $|x - 3| = 6$ , either: - Case 1: $x - 3 = 6$ , so $x = 9$ - Case 2: $x - 3 = -6$ , so $x = -3$ You can verify both solutions work in the original equation. The question asks for the sum of all possible values: $9 + (-3) = 6$ . Looking at the wrong answers: Answer B (9) represents only one of the two solutions—a common error when students forget that absolute value equations typically have two solutions. Answer C (12) might come from adding the absolute values of both solutions ( $|9| + |-3| = 12$ ) instead of their actual sum. Answer D (15) could result from arithmetic errors during the isolation steps or incorrectly calculating $9 + 3 = 12$ and then making another error. The key strategy for absolute value equations is to always consider both the positive and negative cases after isolating the absolute value expression. Remember that $|A| = B$ means $A = B$ or $A = -B$ . Always verify your solutions in the original equation, and read carefully—questions may ask for individual solutions, their sum, or their product.
What is the largest integer solution for $x$ in the inequality $\frac{2x-3}{4} + 6 \ge 2 + \frac{4x}{3}$ ?
Answer: 3 To solve the inequality, first eliminate the fractions by multiplying the entire inequality by the least common denominator of 4 and 3, which is 12. This gives: $12(\frac{2x-3}{4}) + 12(6) \ge 12(2) + 12(\frac{4x}{3})$ , which simplifies to $3(2x-3) + 72 \ge 24 + 4(4x)$ . Distribute the terms: $6x - 9 + 72 \ge 24 + 16x$ . Combine like terms: $6x + 63 \ge 24 + 16x$ . Subtract $6x$ from both sides: $63 \ge 24 + 10x$ . Subtract 24 from both sides: $39 \ge 10x$ . Divide by 10: $3.9 \ge x$ , or $x \le 3.9$ . The question asks for the largest integer solution, which is the largest integer less than or equal to 3.9. That integer is 3.
Solve the inequality $-4(2y-3)\le 8-6y.$
Answer: $y\ge 2$ When solving linear inequalities, you follow the same steps as solving equations, with one crucial exception: flipping the inequality sign when multiplying or dividing by a negative number. Start by distributing the $-4$ on the left side: $-4(2y-3) = -8y + 12$ . The inequality becomes $-8y + 12 \leq 8 - 6y$ . Next, collect like terms by adding $8y$ to both sides: $12 \leq 8 - 6y + 8y$ , which simplifies to $12 \leq 8 + 2y$ . Subtract $8$ from both sides: $4 \leq 2y$ . Finally, divide both sides by $2$ : $2 \leq y$ , or equivalently, $y \geq 2$ . Now let's examine why the other answers are incorrect. Choice B ( $y \leq 2$ ) flips the inequality sign incorrectly—this would happen if you mistakenly thought you needed to flip the sign when dividing by positive 2. Choice C ( $y < -2$ ) and Choice D ( $y > -2$ ) both have the wrong number entirely. These errors likely stem from sign mistakes during the distribution step or when moving terms across the inequality. The correct answer is A: $y \geq 2$ . Study tip: When solving inequalities, double-check your work by testing a value. For $y \geq 2$ , try $y = 3$ : $-4(6-3) = -12$ and $8-18 = -10$ . Since $-12 \leq -10$ is true, your solution checks out. Always remember: only flip the inequality sign when multiplying or dividing by negative numbers.
Solve for $x$ : $\dfrac{7-x}{2}=\dfrac{x+1}{5}.$
Answer: $x=\dfrac{33}{7}$ When you encounter an equation with fractions set equal to each other, the most efficient approach is to cross-multiply. This eliminates the fractions and creates a linear equation you can solve step by step. Starting with $\frac{7-x}{2}=\frac{x+1}{5}$ , cross-multiply by multiplying the numerator of each fraction by the denominator of the other: $5(7-x) = 2(x+1)$ . Distribute on both sides: $35 - 5x = 2x + 2$ . Now collect all terms with $x$ on one side and constants on the other. Subtract $2x$ from both sides: $35 - 7x = 2$ . Subtract 35 from both sides: $-7x = -33$ . Finally, divide by $-7$ : $x = \frac{33}{7}$ . Looking at the answer choices, option A gives $x = \frac{33}{7}$ , which matches our result. Option B shows $x = \frac{7}{33}$ , which would result from incorrectly flipping the fraction at the end. Option C gives $x = -\frac{33}{7}$ , which comes from a sign error when dividing by $-7$ . Option D shows $x = \frac{27}{7}$ , which results from arithmetic errors during the distribution or combining like terms steps. You can always verify your answer by substituting back into the original equation. With $x = \frac{33}{7}$ , both sides equal $-\frac{19}{14}$ , confirming our solution is correct. Study tip: When solving rational equations, always cross-multiply first to clear fractions, then solve the resulting linear equation systematically. Double-check by substituting your answer back into the original equation.

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