Solving One Variable Linear Equations/Inequalities - Algebra 1

Solve for $x$ : $2(x - 3) + 5 = 15$
Answer: $x = 8$ This question tests your ability to solve linear equations in one variable using inverse operations and properties of equality. The properties of equality tell us that we can add, subtract, multiply, or divide the same number on both sides of an equation without changing the solution—think of it like keeping a balance scale level. In this equation, we start by distributing 2 across the parentheses: 2(x - 3) + 5 becomes 2x - 6 + 5. Then we combine like terms to get 2x - 1 = 15, and solving from there gives us 2x = 16, so x = 8. Choice C is correct because it follows all the steps properly: distributes correctly (2x - 6), combines like terms (-6 + 5 = -1), and isolates x accurately to get x = 8. Choice B makes a common mistake: it forgets to distribute 2 to both terms inside the parentheses, calculating 2x - 3 instead of 2x - 6. To check if your answer is right, plug it back into the original equation: substitute your x-value and see if both sides equal the same number. This is a great habit that catches mistakes!
Solve the inequality: $3 - 2(x - 1) > 7 - x$
Answer: $x < -2$ First, expand the left side: $3 - 2(x - 1) = 3 - 2x + 2 = 5 - 2x$ . So the inequality becomes $5 - 2x > 7 - x$ . Subtract 5 from both sides: $-2x > 2 - x$ . Add $x$ to both sides: $-x > 2$ . Multiply both sides by $-1$ (which reverses the inequality): $x < -2$ . Choice B incorrectly forgets to reverse the inequality when multiplying by $-1$ . Choice C results from an arithmetic error. Choice D combines both errors.
Solve for $x$ : $\frac{x}{3} + 2 = 7$
Answer: $x = 15$ This question tests your ability to solve linear equations in one variable using inverse operations and properties of equality. To solve an equation means to find the value of the variable that makes the equation true, and we do this by using inverse operations to isolate the variable on one side while keeping the equation balanced. Starting with $\frac{x}{3} + 2 = 7$ , we first subtract 2 from both sides to get $\frac{x}{3} = 5$ . Then we multiply both sides by 3 to isolate x: $x = 15$ . Choice C is correct because it follows all the steps properly: uses inverse operations and isolates x accurately to get $x = 15$ . Choice B is a close attempt, but it doesn't quite finish the job: it stops at $\frac{x}{3} = 5$ without multiplying both sides by 3 to fully isolate x. Here's a reliable strategy for solving linear equations: (1) simplify each side (distribute and combine like terms), (2) get all x-terms on one side and all numbers on the other side, (3) combine the x-terms and combine the numbers, (4) divide by the coefficient to isolate x. If you follow these steps in order, you'll get the right answer every time!
Solve for $x$ : $3x + 7 = 22$
Answer: $x = 5$ This question tests your ability to solve linear equations in one variable using inverse operations and properties of equality. To solve an equation means to find the value of the variable that makes the equation true, and we do this by using inverse operations to isolate the variable on one side while keeping the equation balanced. Starting with 3x + 7 = 22, we first subtract 7 from both sides to get 3x = 15. Then we divide both sides by 3 to isolate x: x = 15 ÷ 3 = 5. Choice B is correct because it follows all the steps properly: subtracting 7 from both sides and then dividing by 3 to get x = 5. Choice A makes an error when dividing: 15 ÷ 3 equals 5, not 3. Here's a reliable strategy for solving linear equations: (1) simplify each side (distribute and combine like terms), (2) get all x-terms on one side and all numbers on the other side, (3) combine the x-terms and combine the numbers, (4) divide by the coefficient to isolate x. If you follow these steps in order, you'll get the right answer every time!
Solve for $x$ : $4(2x-1)-3(x+2)=17$
Answer: $x=\frac{27}{5}$ This question tests your ability to solve linear equations in one variable using inverse operations and properties of equality. To solve an equation means to find the value of the variable that makes the equation true, and we do this by using inverse operations to isolate the variable on one side while keeping the equation balanced. In this equation, we start by distributing 4 and -3 across the parentheses: 4(2x-1) -3(x+2) becomes 8x-4-3x-6=17, then combine like terms to get 5x-10=17, add 10 to both sides to get 5x=27, and divide by 5 to get x=27/5. Choice C is correct because it follows all the steps properly: distributes correctly, combines like terms, and isolates x accurately to get x=27/5. Choice A makes a common mistake: it forgets to distribute -3 to both terms inside the parentheses, calculating wrong combinations leading to x=22/5 instead of 27/5. Here's a reliable strategy for solving linear equations: (1) simplify each side (distribute and combine like terms), (2) get all x-terms on one side and all numbers on the other side, (3) combine the x-terms and combine the numbers, (4) divide by the coefficient to isolate x. If you follow these steps in order, you'll get the right answer every time!
What is the solution set for $3x - 5 > 16$ ?
Answer: $x > 7$ This question tests your ability to solve linear inequalities in one variable using inverse operations and properties of equality. To solve an inequality means to find all values of the variable that make the inequality true, and we do this by using inverse operations to isolate the variable on one side while keeping the inequality balanced. Starting with 3x - 5 > 16, we first add 5 to both sides to get 3x > 21. Then we divide both sides by 3 to isolate x: x > 21 ÷ 3 = 7. Choice A is correct because it follows all the steps properly: adding 5 to both sides and then dividing by 3 to get x > 7. Choice C makes an error when adding: 16 + 5 equals 21, not 11, so we get 3x > 21 and x > 7, not x > 11/3. The key to avoiding sign errors: when you move a term to the other side of the equation, its sign changes. If you have -5 on the left and want to move it right, it becomes +5 on the right. Think 'move and flip the sign' to keep track!
If $ax + 5 = 3x - 7$ and $a \neq 3$ , solve for $x$ in terms of $a$ .
Answer: $x = \frac{-12}{a - 3}$ Starting with $ax + 5 = 3x - 7$ , subtract $3x$ from both sides: $ax - 3x + 5 = -7$ . Factor out $x$ : $(a - 3)x + 5 = -7$ . Subtract 5 from both sides: $(a - 3)x = -12$ . Divide by $(a - 3)$ : $x = \frac{-12}{a - 3}$ . Choice A has the wrong sign in the numerator. Choice C has the wrong sign in the denominator (note that $\frac{-12}{a-3} = \frac{12}{3-a}$ , but the question asks for the form with $(a-3)$ ). Choice D has the wrong numerator entirely.
Solve for $x$ : $5x - 8 = 2x + 10$
Answer: $x = 6$ This question tests your ability to solve linear equations in one variable using inverse operations and properties of equality. The properties of equality tell us that we can add, subtract, multiply, or divide the same number on both sides of an equation without changing the solution—think of it like keeping a balance scale level. When the variable appears on both sides like $5x - 8 = 2x + 10$ , a good strategy is to move all x-terms to one side (let's move them to the left by subtracting 2x from both sides): $3x - 8 = 10$ . Then move constants to the other side by adding 8 to both sides, giving us $3x = 18$ , so $x = 6$ . Choice B is correct because it follows all the steps properly: combines like terms, and isolates x accurately to get $x = 6$ . Choice C has a sign error—a tricky spot! When moving the constants, if you subtract instead of add, you might end up with a negative value like $x = -6$ . Here's a reliable strategy for solving linear equations: (1) simplify each side (distribute and combine like terms), (2) get all x-terms on one side and all numbers on the other side, (3) combine the x-terms and combine the numbers, (4) divide by the coefficient to isolate x. If you follow these steps in order, you'll get the right answer every time!
What value of $x$ makes the equation true? $\frac{x}{3} + 2 = 7$
Answer: $x = 15$ This question tests your ability to solve linear equations in one variable using inverse operations and properties of equality. To solve an equation means to find the value of the variable that makes the equation true, and we do this by using inverse operations to isolate the variable on one side while keeping the equation balanced. Starting with x/3 + 2 = 7, we first subtract 2 from both sides to get x/3 = 5. Then we multiply both sides by 3 to isolate x: x = 5 × 3 = 15. Choice A is correct because it follows all the steps properly: subtracting 2 from both sides and then multiplying by 3 to get x = 15. Choice B makes a common mistake: it forgets the final step and stops at x/3 = 5, giving x = 3 instead of multiplying both sides by 3. To check if your answer is right, plug it back into the original equation: substitute your x-value and see if both sides equal the same number. This is a great habit that catches mistakes!
Solve the inequality: $-2x + 8 \le 4$
Answer: $x \ge 2$ This question tests your ability to solve linear inequalities in one variable using inverse operations and properties of equality. When solving inequalities, we follow the same steps as equations with one critical rule: if we multiply or divide both sides by a negative number, we must reverse the direction of the inequality symbol (< becomes >, ≤ becomes ≥). Solving -2x + 8 ≤ 4, we first subtract 8 from both sides to get -2x ≤ -4. When we divide both sides by -2, remember the key rule: we must flip the inequality direction! So -2x ≤ -4 becomes x ≥ 2. Choice B is correct because it properly applies the inequality reversal rule when dividing by -2, giving us x ≥ 2. Choice A forgets the critical inequality rule: when dividing both sides by a negative number, we must flip the inequality direction, so ≤ should become ≥. For inequalities, there's just one extra rule to remember beyond regular equation solving: if you multiply or divide by a negative number (like going from -2x ≤ -4 to x ≥ 2), flip the inequality direction. Everything else works exactly the same as equations!
Solve for $x$ : $\frac{3}{4}x - 2 = 1$
Answer: $x = 4$ This question tests your ability to solve linear equations in one variable using inverse operations and properties of equality. To solve an equation means to find the value of the variable that makes the equation true, and we do this by using inverse operations to isolate the variable on one side while keeping the equation balanced. Starting with (3/4)x - 2 = 1, we first add 2 to both sides to get (3/4)x = 3. Finally, we multiply both sides by 4/3 (the reciprocal) to isolate x: x = 3 * (4/3) = 4. Choice A is correct because it follows all the steps properly: adds the constant and multiplies by the reciprocal accurately to get x = 4. Choice B is a close attempt, but it doesn't quite finish the job: it stops at (3/4)x = 3 without multiplying by 4/3, or perhaps divides incorrectly to get x = 4/3. Here's a reliable strategy for solving linear equations: (1) simplify each side (distribute and combine like terms), (2) get all x-terms on one side and all numbers on the other side, (3) combine the x-terms and combine the numbers, (4) divide by the coefficient to isolate x. If you follow these steps in order, you'll get the right answer every time!
For what value of $m$ does the equation $5x + m = 2(x + 4) + 3x$ have infinitely many solutions?
Answer: $m = 8$ First simplify the right side: $2(x + 4) + 3x = 2x + 8 + 3x = 5x + 8$ . So the equation becomes $5x + m = 5x + 8$ . For this equation to have infinitely many solutions, both sides must be identical, which happens when $m = 8$ . When $m = 8$ , we get $5x + 8 = 5x + 8$ , which is always true. Choice B would give $5x - 8 = 5x + 8$ , leading to $-8 = 8$ (no solution). Choice C would give $5x = 5x + 8$ , leading to $0 = 8$ (no solution). Choice D would give $5x + 4 = 5x + 8$ , leading to $4 = 8$ (no solution).
What is the solution set for $3x - 5 > 16$ ?
Answer: $x > 7$ This question tests your ability to solve linear inequalities in one variable using inverse operations and properties of equality. To solve an inequality means to find all values of the variable that make the inequality true, and we do this by using inverse operations to isolate the variable on one side while keeping the inequality balanced. Starting with 3x - 5 > 16, we first add 5 to both sides to get 3x > 21. Then we divide both sides by 3 to isolate x: x > 21 ÷ 3 = 7. Choice A is correct because it follows all the steps properly: adding 5 to both sides and then dividing by 3 to get x > 7. Choice C makes a calculation error: it seems to get x > 11/3, but 21 ÷ 3 = 7, not 11/3. For inequalities, there's just one extra rule to remember beyond regular equation solving: if you multiply or divide by a negative number, flip the inequality direction. Everything else works exactly the same as equations!
Solve for $x$ : $\frac{2x - 1}{4} = 3$
Answer: $x = \frac{13}{2}$ This question tests your ability to solve linear equations in one variable using inverse operations and properties of equality. To solve an equation means to find the value of the variable that makes the equation true, and we do this by using inverse operations to isolate the variable on one side while keeping the equation balanced. Starting with (2x - 1)/4 = 3, we first multiply both sides by 4 to eliminate the denominator, giving us 2x - 1 = 12. Then we add 1 to both sides: 2x = 13. Finally, we divide both sides by 2 to isolate x: x = 13/2. Choice A is correct because it follows all the steps properly: eliminates the fraction correctly and isolates x accurately to get x = 13/2. Choice D makes a common mistake: it forgets to distribute to both terms inside the parentheses, calculating as if it were 2x/4 = 3 to get x = 6 instead of accounting for the -1. Here's a reliable strategy for solving linear equations: (1) simplify each side (distribute and combine like terms), (2) get all x-terms on one side and all numbers on the other side, (3) combine the x-terms and combine the numbers, (4) divide by the coefficient to isolate x. If you follow these steps in order, you'll get the right answer every time!
Solve the inequality: $3x - 5 > 16$
Answer: $x > 7$ This question tests your ability to solve linear inequalities in one variable using inverse operations and properties of equality. When solving inequalities, we follow the same steps as equations with one critical rule: if we multiply or divide both sides by a negative number, we must reverse the direction of the inequality symbol (< becomes >, ≤ becomes ≥). Starting with 3x - 5 > 16, we first add 5 to both sides to get 3x > 21. Then we divide both sides by 3 (which is positive, so no flip): x > 7. Choice A is correct because it follows all the steps properly: adds the constant and divides accurately to get x > 7. Choice D forgets to isolate correctly—a tricky spot! If you subtract instead of add the 5, you might get x < 7. For inequalities, there's just one extra rule to remember beyond regular equation solving: if you multiply or divide by a negative number (like going from -2x < 6 to x > -3), flip the inequality direction. Everything else works exactly the same as equations!
Solve for $x$ : $6x+4=2(3x+2)$
Answer: Infinitely many solutions This question tests your ability to solve linear equations in one variable using inverse operations and properties of equality. A linear equation in one variable has exactly one solution (unless it's a special case), and we find it by systematically undoing the operations around the variable using inverse operations in reverse order. Starting with $6x+4=2(3x+2)$ , we distribute to get $6x+4=6x+4$ , subtract 6x from both sides: $4=4$ , which is always true, so there are infinitely many solutions. Choice D is correct because it recognizes this as an identity after simplifying, where both sides are equal for all x. Choice A makes a common mistake: it might assume no solution if overlooking that $4=4$ is true, but actually it's true for every x. Here's a reliable strategy for solving linear equations: (1) simplify each side (distribute and combine like terms), (2) get all x-terms on one side and all numbers on the other side, (3) combine the x-terms and combine the numbers, (4) divide by the coefficient to isolate x. If you follow these steps in order, you'll get the right answer every time!
Solve the inequality: $-2x+8\le 4$
Answer: $x\ge 2$ This question tests your ability to solve linear inequalities in one variable using inverse operations and properties of equality. When solving inequalities, we follow the same steps as equations with one critical rule: if we multiply or divide both sides by a negative number, we must reverse the direction of the inequality symbol (< becomes >, ≤ becomes ≥). Solving -2x+8≤4, we first subtract 8 from both sides: -2x≤-4. When we divide both sides by -2, remember the key rule: we must flip the inequality direction! So x≥2. Choice B is correct because it follows all the steps properly and flips the inequality when dividing by negative to get x≥2. Choice A forgets the critical inequality rule: when dividing both sides by a negative number, we must flip the inequality direction, so ≤ should be ≥. For inequalities, there's just one extra rule to remember beyond regular equation solving: if you multiply or divide by a negative number (like going from -2x < 6 to x > -3), flip the inequality direction. Everything else works exactly the same as equations!
What is the solution to $3x + 7 = 22$ ?
Answer: $x = 5$ This question tests your ability to solve linear equations in one variable using inverse operations and properties of equality. To solve an equation means to find the value of the variable that makes the equation true, and we do this by using inverse operations to isolate the variable on one side while keeping the equation balanced. Starting with 3x + 7 = 22, we first subtract 7 from both sides to get 3x = 15. Finally, we divide both sides by 3 to isolate x: x = 5. Choice A is correct because it follows all the steps properly: subtracts the constant and divides by the coefficient accurately to get x = 5. Choice B has a sign error—a tricky spot! When subtracting 7, if you add instead, you might get a negative value like x = -5. Here's a reliable strategy for solving linear equations: (1) simplify each side (distribute and combine like terms), (2) get all x-terms on one side and all numbers on the other side, (3) combine the x-terms and combine the numbers, (4) divide by the coefficient to isolate x. If you follow these steps in order, you'll get the right answer every time!
Solve for $x$ : $3x + 7 = 22$
Answer: $x = 5$ This question tests your ability to solve linear equations in one variable using inverse operations and properties of equality. To solve an equation means to find the value of the variable that makes the equation true, and we do this by using inverse operations to isolate the variable on one side while keeping the equation balanced. Starting with 3x + 7 = 22, we first subtract 7 from both sides to get 3x = 15. Then we divide both sides by 3 to isolate x: x = 15 ÷ 3 = 5. Choice B is correct because it follows all the steps properly: subtracting 7 from both sides and then dividing by 3 to get x = 5. Choice C makes a common mistake: it stops at x = 15/3 without simplifying the fraction, which equals 5. Here's a reliable strategy for solving linear equations: (1) simplify each side (distribute and combine like terms), (2) get all x-terms on one side and all numbers on the other side, (3) combine the x-terms and combine the numbers, (4) divide by the coefficient to isolate x. If you follow these steps in order, you'll get the right answer every time!
What is $x$ in the equation $5x - 8 = 2x + 10$ ?
Answer: $x = 6$ This question tests your ability to solve linear equations in one variable using inverse operations and properties of equality. A linear equation in one variable has exactly one solution (unless it's a special case), and we find it by systematically undoing the operations around the variable using inverse operations in reverse order. When the variable appears on both sides like 5x - 8 = 2x + 10, a good strategy is to move all x-terms to one side (let's move them to the left by subtracting 2x from both sides): 3x - 8 = 10. Then move constants to the other side by adding 8: 3x = 18. This gives us 3x = 18, so x = 6. Choice A is correct because it follows all the steps properly: subtracting 2x from both sides, adding 8 to both sides, and dividing by 3 to get x = 6. Choice B has a sign error—a tricky spot! It somehow gets x = -6, but when we check: 5(-6) - 8 = -38 and 2(-6) + 10 = -2, which aren't equal. When solving equations with variables on both sides, you can move the variable to either side—choose whichever keeps your coefficient positive to avoid extra negative signs and reduce errors!

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