Q1. Solve for y: 4y/3 + 5 = 9y/8 + 7

Background

Topic: Solving Linear Equations

This question tests your ability to solve linear equations in one variable by combining like terms and isolating the variable.

Key Terms and Formulas

• Linear equation: An equation of the form .

• To solve, collect all terms with the variable on one side and constants on the other.

Step-by-Step Guidance

1. Identify all terms containing and all constant terms.

2. Subtract from both sides to get all terms on one side.

3. Subtract $5$ from both sides to get all constants on the other side.

4. Find a common denominator to combine the terms.

Try solving on your own before revealing the answer!

Q2. Solve for u: u/7 - 5/3 = -6u/7 + 2

Background

Topic: Solving Linear Equations with Fractions

This question tests your ability to solve linear equations involving fractions and variables on both sides.

Key Terms and Formulas

• Linear equation:

• Clear denominators by multiplying both sides by the least common denominator (LCD).

Step-by-Step Guidance

1. Identify the denominators: 7 and 3.

2. Multiply both sides by the LCD (which is 21) to eliminate fractions.

3. Distribute and simplify each term.

4. Collect all terms on one side and constants on the other.

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Q3. A Web music store offers two versions of a song. The standard version is 2.6 MB, the high-quality version is 4.8 MB. Yesterday, there were 1010 downloads for a total of 4210 MB. How many downloads of the standard version were there?

Background

Topic: Systems of Linear Equations (Word Problems)

This question tests your ability to set up and solve a system of equations based on a real-world scenario.

Key Terms and Formulas

• Let = number of standard downloads, = number of high-quality downloads.

• Total downloads:

• Total size:

Step-by-Step Guidance

1. Let be the number of standard downloads and the number of high-quality downloads.

2. Write the two equations: and .

3. Solve the first equation for in terms of .

4. Substitute this expression for into the second equation and solve for .

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Q4. To rent a meeting room, a college charges a t$ they could rent the room?

Background

Topic: Linear Inequalities

This question tests your ability to set up and solve inequalities based on a real-world scenario.

Key Terms and Formulas

• Cost function:

• "At most" means

Step-by-Step Guidance

1. Write the inequality: .

2. Subtract 34 from both sides to isolate the term with .

3. Divide both sides by 6.40 to solve for .

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Q5. Jina is renting a truck for one day. Company A charges m$ will Company A charge less than Company B?

Background

Topic: Linear Inequalities (Comparing Costs)

This question tests your ability to compare two linear cost functions and solve an inequality.

Key Terms and Formulas

• Company A:

• Company B:

• Set up the inequality:

Step-by-Step Guidance

1. Write the inequality: .

2. Subtract from both sides to get all terms on one side.

3. Subtract 30 from both sides to get constants on the other side.

4. Divide both sides by 0.20 to solve for .

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Q6. Graph the solution to the inequality: or on the number line.

Background

Topic: Graphing Solutions to Inequalities

This question tests your ability to represent the solution set of an inequality on a number line.

Key Terms and Formulas

• "Or" means the union of two intervals.

• Open circles indicate endpoints not included.

Step-by-Step Guidance

1. Identify the two intervals: and .

2. Draw a number line and mark and $7$.

3. Shade the regions to the right of and to the left of $7$.

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Q7. Solve:

Background

Topic: Solving Linear Inequalities in Two Variables

This question tests your ability to solve and graph inequalities involving two variables.

Key Terms and Formulas

• Linear inequality:

• To graph, solve for in terms of .

Step-by-Step Guidance

1. Multiply both sides by 2 to clear the denominator.

2. Express the inequality in the form .

3. Recognize that this represents the region above a certain line in the -plane.

Try solving on your own before revealing the answer!