Math: Algebraic Concepts Flashcards

A plumber charges a $75 service fee for a visit, plus an additional $50 per hour of work. Which equation correctly represents the total cost, C, for a job that takes h hours to complete?

Answer: C = 50h + 75

The total cost is the sum of the fixed service fee and the variable hourly charge. The service fee is a one-time charge of $75. The hourly charge is $50 multiplied by the number of hours, h, which is represented as 50h. Therefore, the correct equation is C = 50h + 75.

Which of the following is the simplified form of the expression 3(x - 4) + 2(x + 5)?

Answer: 5x - 2

First, use the distributive property to eliminate the parentheses: 3(x - 4) becomes 3x - 12, and 2(x + 5) becomes 2x + 10. The expression is now 3x - 12 + 2x + 10. Next, combine like terms: combine the x-terms (3x + 2x = 5x) and the constant terms (-12 + 10 = -2). The simplified expression is 5x - 2.

What is the value of the expression 4a² - 2b when a = 3 and b = 5?

Answer: 26

Substitute the given values into the expression: 4(3)² - 2(5). Following the order of operations (PEMDAS), first evaluate the exponent: 3² = 9. The expression becomes 4(9) - 2(5). Next, perform the multiplications: 4(9) = 36 and 2(5) = 10. The expression is now 36 - 10. Finally, perform the subtraction: 36 - 10 = 26.

Which of the following is the correct solution for x in the equation 5x - 8 = 2x + 7?

Answer: x = 5

To solve for x, first move the variable terms to one side. Subtract 2x from both sides: 5x - 2x - 8 = 7, which simplifies to 3x - 8 = 7. Next, move the constant terms to the other side by adding 8 to both sides: 3x = 7 + 8, which simplifies to 3x = 15. Finally, divide both sides by 3 to isolate x: x = 15 / 3, so x = 5.

A student is solving the inequality 12 - 3x > 6. Which of the following represents the correct solution?

Answer: x < 2

First, subtract 12 from both sides of the inequality: -3x > 6 - 12, which simplifies to -3x > -6. Next, divide both sides by -3. When multiplying or dividing both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. So, '>' becomes '<'. The solution is x < (-6 / -3), which simplifies to x < 2.

A recipe for a batch of cookies requires 2 cups of flour for every 3/4 cup of sugar. If a baker uses 6 cups of flour, how much sugar is needed?

Answer: 2.25 cups

This is a proportion problem. Set up a ratio of flour to sugar: (2 cups flour) / (0.75 cups sugar). Set this equal to the ratio for the new batch: (6 cups flour) / (x cups sugar). The proportion is 2/0.75 = 6/x. To solve, cross-multiply: 2 * x = 6 * 0.75. This gives 2x = 4.5. Divide both sides by 2 to find x = 2.25 cups of sugar.