The problem asks to find x from the phone plan equation $18 + 7x = 46$, where 18 is the base charge and $7x$ represents 7 gigabytes at x dollars each. Subtract 18 from both sides: $7x = 46 - 18 = 28$. Divide both sides by 7: $x = 28 \div 7 = 4$. A common mistake would be dividing the wrong terms or making arithmetic errors in subtraction. Always isolate the variable term first, then divide by its coefficient.

The problem asks to solve (2x - 1)/3 = (x + 5)/2 for x. Cross-multiply to get 2(2x - 1) = 3(x + 5), which gives 4x - 2 = 3x + 15. Subtract 3x from both sides: x - 2 = 15, then add 2: x = 17. A common error is making mistakes during cross-multiplication or not properly distributing. When solving rational equations, cross-multiplication is often the most efficient method to eliminate fractions.

The problem describes a lab mixture equation x + 12 = 45, where x is the initial amount and 12 mL is added to get 45 mL total. Subtract 12 from both sides: x = 45 - 12 = 33. A common mistake would be adding 12 instead of subtracting or making arithmetic errors. This is a simple one-step equation requiring only subtraction to isolate the variable.

The problem asks to solve 7x - 3 = 2x + 17/2 for x. Subtract 2x from both sides: 5x - 3 = 17/2, then add 3: 5x = 17/2 + 6/2 = 23/2. Finally, divide by 5: x = 23/10. A common error is incorrect fraction arithmetic when combining terms. When adding fractions to whole numbers, convert the whole number to a fraction with the same denominator first.

The problem describes a streaming service billing equation 9 + 5x = 24, where 9 is the base fee and 5x represents 5 movies at x dollars each. Subtract 9 from both sides: 5x = 15, then divide by 5: x = 3. A common mistake would be dividing by the wrong coefficient or making arithmetic errors in subtraction. When solving real-world problems, identify the fixed and variable costs before isolating the variable.

We need to solve $12 + 0.75x = 0.5x + 18$ for the number of gigabytes $x$. Subtracting $0.5x$ from both sides: $12 + 0.75x - 0.5x = 18$, which simplifies to $12 + 0.25x = 18$. Subtracting 12 from both sides: $0.25x = 6$. Dividing by 0.25: $x = 6 ÷ 0.25 = 24$. A common error is making arithmetic mistakes with decimals or incorrectly combining the $x$ terms. When working with decimal coefficients, consider converting to fractions to avoid calculation errors.

This problem asks us to find the monthly fee $x$ when the total cost after 6 months equals $18 + 6x = 3x + 54$. To solve, we first subtract $3x$ from both sides: $18 + 6x - 3x = 54$, which simplifies to $18 + 3x = 54$. Next, we subtract 18 from both sides: $3x = 54 - 18 = 36$. Finally, dividing both sides by 3 gives us $x = 12$. A common error is incorrectly combining the $x$ terms or making arithmetic mistakes when subtracting. When solving equations with variables on both sides, always collect like terms on one side first.

When you encounter an equation with fractions, your goal is to clear the denominators and solve for the variable systematically.

To solve $$\frac{x+1}{2} + \frac{x+2}{3} = \frac{16}{3}$$, find a common denominator for the left side. The LCD of 2 and 3 is 6, so multiply each term appropriately:

$$\frac{3(x+1)}{6} + \frac{2(x+2)}{6} = \frac{16}{3}$$

This simplifies to:

$$\frac{3x+3+2x+4}{6} = \frac{16}{3}$$

$$\frac{5x+7}{6} = \frac{16}{3}$$

Cross-multiply to eliminate fractions:

$$3(5x+7) = 6 \cdot 16$$

$$15x + 21 = 96$$

$$15x = 75$$

$$x = 5$$

You can verify: $$\frac{5+1}{2} + \frac{5+2}{3} = \frac{6}{2} + \frac{7}{3} = 3 + \frac{7}{3} = \frac{16}{3}$$ ✓

Choice A ($$-\frac{1}{3}$$) likely results from sign errors during fraction manipulation. Choice B (1) might come from incorrectly combining like terms or making arithmetic mistakes when finding the common denominator. Choice D (25) could result from errors in cross-multiplication, such as multiplying $$15x = 75$$ incorrectly or confusing the final division step.

Strategy tip: When solving equations with multiple fractions, always find a common denominator first, then cross-multiply to eliminate fractions entirely. This reduces the chance of arithmetic errors and makes the algebra more straightforward. Always substitute your answer back into the original equation to verify it works.

This equation requires careful algebraic manipulation, particularly when dealing with nested parentheses. The key is to work systematically from the inside out, then combine like terms to isolate the variable.

Start with the left side: $$3(x - 2(x + 1))$$. First, distribute the $$-2$$ inside the inner parentheses: $$x - 2(x + 1) = x - 2x - 2 = -x - 2$$. Now distribute the $$3$$: $$3(-x - 2) = -3x - 6$$.

For the right side: $$4(x - 5) + 2 = 4x - 20 + 2 = 4x - 18$$.

The equation becomes: $$-3x - 6 = 4x - 18$$

Collect like terms by adding $$3x$$ to both sides: $$-6 = 7x - 18$$

Add $$18$$ to both sides: $$12 = 7x$$

Therefore: $$x = \frac{12}{7}$$

Choice A (3) likely comes from making errors in the distribution process and getting a simpler equation. Choice B ($$\frac{16}{7}$$) probably results from a sign error when combining the constant terms—getting $$16 = 7x$$ instead of $$12 = 7x$$. Choice C ($$\frac{16}{3}$$) suggests confusion in both the coefficient of $$x$$ and the constant term, possibly mixing up the $$7$$ and $$3$$ from the distribution steps.

When solving multi-step equations with nested parentheses, always distribute from the innermost parentheses outward, then carefully track positive and negative signs as you combine like terms. Double-check by substituting your answer back into the original equation.

This problem tests your ability to set up and solve profit equations. When you see questions about company costs, revenues, and profits, remember that profit equals revenue minus costs.

First, let's establish the key relationships. The cost function is $$C(x) = 15x + 500$$, where $$x$$ is the number of units produced. Since each unit sells for $25, the revenue function is $$R(x) = 25x$$. Profit is calculated as $$P(x) = R(x) - C(x) = 25x - (15x + 500) = 10x - 500$$.

To find when profit equals $4,500, set up the equation: $$10x - 500 = 4500$$. Adding 500 to both sides gives $$10x = 5000$$, so $$x = 500$$. This confirms answer choice D is correct.

Let's examine why the other answers are wrong. Choice A (300 units) would yield a profit of $$10(300) - 500 = 2500$$, which is $2,000 short of the target. Choice B (400 units) gives $$10(400) - 500 = 3500$$, still $1,000 below the goal. Choice C (450 units) results in $$10(450) - 500 = 4000$$, which is $500 less than needed.

Each incorrect answer represents a common calculation error: either mistakes in setting up the profit equation or arithmetic errors when solving.

Study tip: Always write out the profit equation explicitly as Revenue - Costs before substituting numbers. This prevents confusion about which values represent income versus expenses, and double-check your arithmetic by substituting your answer back into the original profit equation.