Step-by-Step Guidance for Solving Rational and Linear Equations (Intermediate Algebra)

Q72. Solve for x: \( \frac{3x+2}{28} = 3 \)

Background

Topic: Solving Linear Equations with Rational Expressions

This question tests your ability to solve equations involving rational expressions by isolating the variable.

Key Terms and Formulas:

• Rational Expression: An expression that involves fractions with variables in the numerator, denominator, or both.

• Linear Equation: An equation of the first degree (variable raised to the power of 1).

Step-by-Step Guidance

1. Identify the equation: \( \frac{3x+2}{28} = 3 \).

2. To eliminate the denominator, multiply both sides of the equation by 28.

3. After multiplying, rewrite the equation so the variable term is isolated.

4. Set up the equation to solve for x by moving constants to the other side.

Try solving on your own before revealing the answer!

Final Answer: x = 26

Multiplying both sides by 28 gives \( 3x+2 = 84 \). Subtract 2 from both sides to get \( 3x = 82 \), then divide by 3 to find \( x = 26 \).

Q73. Solve for x: \( \frac{24-3x}{12} = 1-4 \)

Background

Topic: Solving Linear Equations with Rational Expressions

This question tests your ability to manipulate rational expressions and solve for the variable.

Key Terms and Formulas:

• Rational Expression

• Linear Equation

Step-by-Step Guidance

1. First, simplify the right side: \( 1-4 = -3 \).

2. Rewrite the equation: \( \frac{24-3x}{12} = -3 \).

3. Multiply both sides by 12 to clear the denominator.

4. Isolate the variable term and set up the equation to solve for x.

Try solving on your own before revealing the answer!

Final Answer: x = 11

Multiplying both sides by 12 gives \( 24-3x = -36 \). Subtract 24 from both sides to get \( -3x = -60 \), then divide by -3 to find \( x = 20 \).

Q74. Solve for x: \( \frac{3x+9}{5} = 12 \)

Background

Topic: Solving Linear Equations with Rational Expressions

This question tests your ability to solve for x in equations involving fractions.

Key Terms and Formulas:

• Rational Expression

• Linear Equation

Step-by-Step Guidance

1. Multiply both sides by 5 to eliminate the denominator.

2. Rewrite the equation so the variable term is isolated.

3. Move constants to the other side to set up for solving x.

Try solving on your own before revealing the answer!

Final Answer: x = 1

Multiplying both sides by 5 gives \( 3x+9 = 60 \). Subtract 9 from both sides to get \( 3x = 51 \), then divide by 3 to find \( x = 17 \).

Q75. Solve for x: \( \frac{2x-1}{x+3} = 9 \)

Background

Topic: Solving Rational Equations

This question tests your ability to solve equations where the variable appears in both the numerator and denominator.

Key Terms and Formulas:

• Rational Equation: An equation involving fractions with variables in the denominator.

Step-by-Step Guidance

1. Multiply both sides by \( x+3 \) to clear the denominator.

2. Expand and rewrite the equation so all terms are on one side.

3. Set up the equation to solve for x.

Try solving on your own before revealing the answer!

Final Answer: x = 2

Multiplying both sides by \( x+3 \) gives \( 2x-1 = 9(x+3) \). Expand to \( 2x-1 = 9x+27 \), then solve for x.

Q76. Solve for x: \( \frac{x+1}{x+4} = 3 \)

Background

Topic: Solving Rational Equations

This question tests your ability to solve equations with variables in both the numerator and denominator.

Key Terms and Formulas:

• Rational Equation

Step-by-Step Guidance

1. Multiply both sides by \( x+4 \) to clear the denominator.

2. Expand and rewrite the equation so all terms are on one side.

3. Set up the equation to solve for x.

Try solving on your own before revealing the answer!

Final Answer: x = 2

Multiplying both sides by \( x+4 \) gives \( x+1 = 3(x+4) \). Expand to \( x+1 = 3x+12 \), then solve for x.