College Algebra Test 2 Review: Step-by-Step Study Guidance

주석 실무 비디오 수업

스터디 가이드 - 스마트 노트

자료에 맞춘 맞춤형 노트, 핵심 정의, 예시, 맥락을 확장해 제공합니다.

Q1. Solve the inequality and write the solution set in interval notation:

Background

Topic: Linear Inequalities

This question tests your ability to solve linear inequalities and express the solution in interval notation.

Key Terms and Formulas

Linear inequality: An inequality involving a linear expression.
Interval notation: A way to describe a set of numbers between two endpoints.

Step-by-Step Guidance

Combine like terms on the left side: simplifies to .
Expand the right side: becomes .
Set up the simplified inequality: .
Subtract from both sides to get all terms on one side.
Subtract $3x$ term.

Try solving on your own before revealing the answer!

Final Answer:

After isolating , you get , which in interval notation is .

Q2. Solve the inequality and write the solution set in interval notation:

Background

Topic: Linear Inequalities

This question checks your understanding of solving inequalities where the variable terms cancel out.

Key Terms and Formulas

Linear inequality: An inequality involving a linear expression.
Interval notation: Describes all possible solutions.

Step-by-Step Guidance

Subtract from both sides to eliminate the variable terms.
Observe what remains: .
Decide if this statement is always true, always false, or sometimes true.

Try solving on your own before revealing the answer!

Final Answer:

Since is always true, the solution is all real numbers, or .

Q3. Solve the compound inequality and write the solution set in interval notation:

Background

Topic: Compound Inequalities

This question tests your ability to solve compound inequalities and express the solution in interval notation.

Key Terms and Formulas

Compound inequality: Two inequalities joined by 'and' (both must be true).
Interval notation: Used to express the solution set.

Step-by-Step Guidance

Break the compound inequality into two parts: and .
Solve each inequality separately for .
For the first part, subtract $4.
For the second part, subtract $4.
Combine the two solution intervals to find the intersection.

Try solving on your own before revealing the answer!

Final Answer:

The solution is the intersection of the two intervals, which is .

Q4. Solve the compound inequality and write the solution set in interval notation:

Background

Topic: Compound Inequalities

This question tests your ability to solve and interpret compound inequalities.

Key Terms and Formulas

Compound inequality: Two inequalities joined by 'and'.
Interval notation: Used to express the solution set.

Step-by-Step Guidance

Interpret the compound inequality: and .
Solve for .
Note that is always true, so focus on the first part.
After isolating , write the solution in interval notation.

Try solving on your own before revealing the answer!

Final Answer:

Solving gives , so the solution is .

Q5. Solve the quadratic inequality and write the solution set in interval notation:

Background

Topic: Quadratic Inequalities

This question tests your ability to solve quadratic inequalities and express the solution in interval notation.

Key Terms and Formulas

Quadratic inequality: An inequality involving a quadratic expression.
Factoring: Expressing the quadratic as a product of binomials.
Test intervals: Checking the sign of the expression in each interval determined by the roots.

Step-by-Step Guidance

Set the quadratic equal to zero: and factor it.
Find the roots of the equation.
Use the roots to divide the number line into intervals.
Test a value from each interval in the original inequality to determine where the expression is positive.
Write the solution set in interval notation, excluding the roots since the inequality is strict ().

Try solving on your own before revealing the answer!

Final Answer:

The solution is all values less than or greater than $6$.

Q6. Solve the quadratic inequality and write the solution set in interval notation:

Background

Topic: Quadratic Inequalities

This question tests your ability to solve quadratic inequalities and express the solution in interval notation.

Key Terms and Formulas

Quadratic inequality: An inequality involving a quadratic expression.
Factoring or quadratic formula: Used to find the roots.
Test intervals: Used to determine where the inequality holds.

Step-by-Step Guidance

Rewrite the inequality as .
Factor the quadratic or use the quadratic formula to find the roots.
Use the roots to divide the number line into intervals.
Test a value from each interval in the original inequality to determine where the expression is less than or equal to zero.
Include the roots in the solution set since the inequality is .

Try solving on your own before revealing the answer!

Final Answer:

The solution is the interval between the two roots, inclusive.

Q7. Solve the rational inequality and write the solution set in interval notation:

Background

Topic: Rational Inequalities

This question tests your ability to solve inequalities involving rational expressions.

Key Terms and Formulas

Rational inequality: An inequality involving a ratio of polynomials.
Critical points: Values where the numerator or denominator is zero.
Test intervals: Used to determine where the inequality holds.

Step-by-Step Guidance

Set the numerator and denominator equal to zero to find critical points.
Use these points to divide the number line into intervals.
Test a value from each interval in the original inequality to determine where the expression is less than or equal to zero.
Remember to exclude values that make the denominator zero from the solution set.

Try solving on your own before revealing the answer!

Final Answer:

The solution is all between (not included) and $3$ (included).

Q8. Solve the rational inequality and write the solution set in interval notation:

Background

Topic: Rational Inequalities

This question tests your ability to solve inequalities involving rational expressions and to express the solution in interval notation.

Key Terms and Formulas

Rational inequality: An inequality involving a ratio of polynomials.
Critical points: Values where the numerator or denominator is zero, or where the expression equals the boundary value.
Test intervals: Used to determine where the inequality holds.

Step-by-Step Guidance

Subtract $5\dfrac{30}{x + 6} - 5 \leq 0$.
Combine into a single rational expression: .
Simplify the numerator and find the critical points by setting the numerator and denominator to zero.
Use these points to divide the number line into intervals.
Test a value from each interval in the original inequality to determine where the expression is less than or equal to zero.

Try solving on your own before revealing the answer!

Final Answer:

The solution is all less than and all greater than or equal to $0$.

Q9. Suppose the velocity of an object is given by , where is time in seconds. Find the intervals where the velocity is negative.

Background

Topic: Quadratic Functions and Inequalities

This question tests your ability to analyze a quadratic function and determine where its values are negative.

Key Terms and Formulas

Quadratic function:
Roots: Values of where
Test intervals: Used to determine where the function is negative.

Step-by-Step Guidance

Set and solve for using the quadratic formula.
Find the two roots, which will divide the -axis into three intervals.
Test a value from each interval in the original function to determine where is negative.
Write the interval(s) where is negative.

Try solving on your own before revealing the answer!

Final Answer: Between sec and sec

The velocity is negative for values between the two roots.

Q10. Solve the equation:

Background

Topic: Linear Equations

This question tests your ability to solve a simple linear equation for .

Key Terms and Formulas

Linear equation: An equation of the form .

Step-by-Step Guidance

Subtract $8x$ term.
Divide both sides by to solve for .

Try solving on your own before revealing the answer!

Final Answer:

Solving gives or .