Algebra: Linear Equations/Inequalities Flashcards

A plumber charges a flat fee of $75 for a service call, plus an additional $45 per hour for labor. The total cost for a service call, C, can be represented by the equation C = 45h + 75, where h is the number of hours of labor. If a customer's total bill was $255, how many hours of labor were required?

Answer: 4 hours

To find the number of hours, substitute the total cost ($255) into the equation for C and solve for h. The equation is 255 = 45h + 75. First, subtract 75 from both sides: 255 - 75 = 45h, which simplifies to 180 = 45h. Then, divide both sides by 45: h = 180 / 45. This gives h = 4. Therefore, the plumber worked for 4 hours.

Which of the following inequalities represents the statement: 'The sum of a number, x, and 8 is at least 20'?

Answer: x + 8 ≥ 20

The phrase 'at least' means 'greater than or equal to'. Therefore, the sum of the number x and 8 must be greater than or equal to 20. This translates to the inequality x + 8 ≥ 20.

A student is selling candy bars for a fundraiser. They want to raise more than $150. They sell small candy bars for $1.50 each and large candy bars for $2.50 each. If 's' represents the number of small candy bars and 'l' represents the number of large candy bars sold, which inequality models this situation?

Answer: 1.50s + 2.50l > 150

The total amount of money raised is the sum of the earnings from small and large candy bars. The earnings from small bars are 1.50s and from large bars are 2.50l. The student wants to raise 'more than' $150, which means the total must be strictly greater than 150. Therefore, the correct inequality is 1.50s + 2.50l > 150.

What is the solution to the equation 3(x - 4) + 2 = 2x + 5?

Answer: x = 15

To solve the equation, first distribute the 3 on the left side: 3x - 12 + 2 = 2x + 5. Combine like terms on the left side: 3x - 10 = 2x + 5. Next, subtract 2x from both sides: x - 10 = 5. Finally, add 10 to both sides to isolate x: x = 15.

A school is organizing a trip. A bus can hold at most 50 students. Which inequality correctly represents the number of students, s, that can be on the bus?

Answer: s ≤ 50

The phrase 'at most' means the number of students can be 50 or any number less than 50. Therefore, the number of students, s, must be less than or equal to 50. This is represented by the inequality s ≤ 50.

Solve the compound inequality: -7 ≤ 2x - 3 < 5

Answer: -2 ≤ x < 4

To solve a compound inequality, perform the same operation on all three parts to isolate the variable in the middle. First, add 3 to all three parts: -7 + 3 ≤ 2x - 3 + 3 < 5 + 3, which simplifies to -4 ≤ 2x < 8. Next, divide all three parts by 2: -4/2 ≤ 2x/2 < 8/2, which simplifies to -2 ≤ x < 4.