Q1. Solve the equation: \( \frac{1}{4}(8x - 12) = 6x \)

Background

Topic: Linear Equations

This question tests your ability to solve linear equations involving distribution and combining like terms.

Key Terms and Formulas

• Distributive Property: \( a(b + c) = ab + ac \)

• Isolate the variable by performing inverse operations.

Step-by-Step Guidance

1. Apply the distributive property to \( \frac{1}{4}(8x - 12) \) to eliminate the parentheses.

2. Write the equation without fractions if possible by multiplying both sides by 4 to clear denominators.

3. Combine like terms and move all terms involving \( x \) to one side of the equation.

4. Move constant terms to the other side and prepare to solve for \( x \).

Try solving on your own before revealing the answer!

Q2. Solve the equation: \( -\frac{1}{7}(x + 14) + \frac{1}{4}(x + 4) = x - 2 \)

Background

Topic: Linear Equations with Fractions

This question tests your ability to solve linear equations that include fractions and require finding a common denominator.

Key Terms and Formulas

• Common Denominator: Used to combine fractional terms.

• Distributive Property: \( a(b + c) = ab + ac \)

Step-by-Step Guidance

1. Distribute the fractions across the parentheses: \( -\frac{1}{7}(x + 14) \) and \( \frac{1}{4}(x + 4) \).

2. Combine the fractional terms on the left side into a single expression.

3. Multiply both sides by the least common denominator (LCD) to clear fractions.

4. Collect like terms and move all \( x \) terms to one side and constants to the other.

Try solving on your own before revealing the answer!

Q3. Solve the equation: \( \frac{x + 7}{5} + \frac{x - 1}{2} = 8 \)

Background

Topic: Linear Equations with Fractions

This question tests your ability to solve equations with multiple fractional terms by finding a common denominator.

Key Terms and Formulas

• Common Denominator: Used to combine fractions.

• Isolate the variable by performing inverse operations.

Step-by-Step Guidance

1. Find the least common denominator (LCD) for the fractions (5 and 2).

2. Multiply both sides of the equation by the LCD to eliminate denominators.

3. Expand and combine like terms to form a linear equation in \( x \).

4. Move all terms involving \( x \) to one side and constants to the other.

Try solving on your own before revealing the answer!

Q4. The three most prominent buildings in a city have a total height of 1800 feet. Jefferson Square Tower is three times as tall as Lincoln Galleria, and Washington Center is 450 feet taller than Lincoln Galleria. Find the height of each building.

Background

Topic: Systems of Linear Equations (Word Problems)

This question tests your ability to translate a word problem into a system of equations and solve for unknowns.

Key Terms and Formulas

• Let variables represent the unknown heights.

• Set up equations based on the relationships described.

Step-by-Step Guidance

1. Let \( x \) represent the height of Lincoln Galleria.

2. Express the height of Jefferson Square Tower as \( 3x \).

3. Express the height of Washington Center as \( x + 450 \).

4. Set up the equation: \( x + 3x + (x + 450) = 1800 \).

5. Combine like terms and prepare to solve for \( x \).

Try solving on your own before revealing the answer!

Q5. A publisher printed 90 million pages last year, which is 125% of the previous year's total. How many pages were printed the previous year? (Round to the nearest hundredth million if necessary.)

Background

Topic: Percent Increase/Decrease

This question tests your ability to work with percentages and solve for an original amount given a percent increase.

Key Terms and Formulas

• Percent Increase Formula: \( \text{New Amount} = \text{Original Amount} \times (1 + \frac{\text{Percent Increase}}{100}) \)

• To find the original amount, rearrange the formula.

Step-by-Step Guidance

1. Let \( x \) be the number of pages printed the previous year.

2. Set up the equation: \( 90 = x \times 1.25 \).

3. Rearrange the equation to solve for \( x \).

4. Divide both sides by 1.25 to isolate \( x \).

Try solving on your own before revealing the answer!

Q6. The sum of three consecutive even integers is 330. Find the integers.

Background

Topic: Consecutive Integer Problems

This question tests your ability to set up and solve equations involving consecutive even integers.

Key Terms and Formulas

• Let the first integer be \( x \), the next is \( x + 2 \), and the third is \( x + 4 \).

• Set up the equation: \( x + (x + 2) + (x + 4) = 330 \).

Step-by-Step Guidance

1. Let \( x \) represent the smallest even integer.

2. Write expressions for the next two consecutive even integers.

3. Set up the equation for their sum equal to 330.

4. Combine like terms and prepare to solve for \( x \).

Try solving on your own before revealing the answer!