Podcast
Questions and Answers
What type of linear equation is x + 1 = x + 1?
What type of linear equation is x + 1 = x + 1?
identity
What would a rational equation be multiplied by to clear the equation of fractions?
What would a rational equation be multiplied by to clear the equation of fractions?
least common denominator
What type of number contains both a real part and an imaginary part, for example, 7 − 3i?
What type of number contains both a real part and an imaginary part, for example, 7 − 3i?
complex
Rewrite 2x + 4x^2 = 9 in standard form.
Rewrite 2x + 4x^2 = 9 in standard form.
What is the imaginary form of √-16?
What is the imaginary form of √-16?
Which of the following correctly states the number of solutions for |3x + 20| > -12?
Which of the following correctly states the number of solutions for |3x + 20| > -12?
Which of the following correctly states the types of real solutions for the discriminant 16?
Which of the following correctly states the types of real solutions for the discriminant 16?
Which of the choices correctly translates the following problem? "The area of a yard is 24 square feet. If the width is 3 feet less than the length, state the dimensions of the yard."
Which of the choices correctly translates the following problem? "The area of a yard is 24 square feet. If the width is 3 feet less than the length, state the dimensions of the yard."
What type of interval best describes -2 ≤ x < 8?
What type of interval best describes -2 ≤ x < 8?
Match the following equations with their solutions:
Match the following equations with their solutions:
What are the values for a, b, and c in the equation -2x = -3x^2 + 11?
What are the values for a, b, and c in the equation -2x = -3x^2 + 11?
When simplifying the expression 3i/(2 - i), by which of the following would you multiply both numerator and denominator?
When simplifying the expression 3i/(2 - i), by which of the following would you multiply both numerator and denominator?
Solve for x: 3x − 4 = 8.
Solve for x: 3x − 4 = 8.
Solve for x: ax = 2x + 5.
Solve for x: ax = 2x + 5.
Solve for x: 2|x + 2| = 4.
Solve for x: 2|x + 2| = 4.
Solve for x: x^2 − 3x − 10 = 0.
Solve for x: x^2 − 3x − 10 = 0.
Solve for x: x^2 − 15 = 1.
Solve for x: x^2 − 15 = 1.
Solve for x: x^2 + 8x + 4 = 0.
Solve for x: x^2 + 8x + 4 = 0.
Solve for x: x^2 + 3x + 8 = 0.
Solve for x: x^2 + 3x + 8 = 0.
Solve the inequality: -8x < 8.
Solve the inequality: -8x < 8.
Solve the inequality: 4 ≤ 2x ≤ 10.
Solve the inequality: 4 ≤ 2x ≤ 10.
Solve the inequality: |x + 1| > 3.
Solve the inequality: |x + 1| > 3.
What is the value of i^15?
What is the value of i^15?
Simplify the expression: (2 + 7i) − (4 − 8i).
Simplify the expression: (2 + 7i) − (4 − 8i).
Multiply (6 − 2i) (1 + i).
Multiply (6 − 2i) (1 + i).
Simplify: 3/(6+i).
Simplify: 3/(6+i).
Solve the equation: 1/2z + 3z = 5/3z − 11.
Solve the equation: 1/2z + 3z = 5/3z − 11.
Solve for x: 3/x - 2 = 8/x + 6.
Solve for x: 3/x - 2 = 8/x + 6.
Solve the equation: 2/x = 3/x+1 + 2/x+1.
Solve the equation: 2/x = 3/x+1 + 2/x+1.
Solve for x: √3x + 1 = 4.
Solve for x: √3x + 1 = 4.
Solve for x: √5x + 5 = √4x + 1.
Solve for x: √5x + 5 = √4x + 1.
The denominator of a fraction is 4 more than its numerator. If the fraction equals 1/3, what is the fraction?
The denominator of a fraction is 4 more than its numerator. If the fraction equals 1/3, what is the fraction?
Flashcards
Identity
Identity
An equation that is always true, regardless of the value of the variable.
Least Common Denominator (LCD)
Least Common Denominator (LCD)
The smallest multiple that is common to all denominators in a set of fractions; used to clear fractions from an equation.
Complex Number
Complex Number
Numbers that include both a real part and an imaginary part (a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit).
Standard Form of a Quadratic Equation
Standard Form of a Quadratic Equation
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Rationalizing the Denominator (Complex Numbers)
Rationalizing the Denominator (Complex Numbers)
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Solution for ax = 2x + 5
Solution for ax = 2x + 5
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Solutions for 2|x + 2| = 4
Solutions for 2|x + 2| = 4
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Solutions for x² − 3x − 10 = 0
Solutions for x² − 3x − 10 = 0
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Solutions for x² − 15 = 1
Solutions for x² − 15 = 1
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Solution to -8x < 8
Solution to -8x < 8
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Solution to 4 ≤ 2x ≤ 10
Solution to 4 ≤ 2x ≤ 10
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Solution to |x + 1| > 3
Solution to |x + 1| > 3
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i^15 Simplification
i^15 Simplification
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(2 + 7i) − (4 − 8i) Result
(2 + 7i) − (4 − 8i) Result
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(6 − 2i)(1 + i) Result
(6 − 2i)(1 + i) Result
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Simplification of 3/(6+i)
Simplification of 3/(6+i)
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Solution for 1/2z + 3z = 5/3z - 11
Solution for 1/2z + 3z = 5/3z - 11
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Solution to 3/x - 2 = 8/x + 6
Solution to 3/x - 2 = 8/x + 6
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Solution to 2/x = 3/(x+1) + 2/(x+1)
Solution to 2/x = 3/(x+1) + 2/(x+1)
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Solution to √3x + 1 = 4
Solution to √3x + 1 = 4
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Solution to √5x + 5 = √4x + 1
Solution to √5x + 5 = √4x + 1
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Fraction Problem
Fraction Problem
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Study Notes
Linear Equations and Solutions
- An equation like x + 1 = x + 1 is classified as an identity.
- The least common denominator is used to eliminate fractions in rational equations.
- Solutions to absolute value inequalities, such as |3x + 20| > -12, encompass all real numbers.
Complex Numbers
- A number with both real and imaginary parts, exemplified by 7 − 3i, is termed complex.
Quadratic Equations
- The standard form of the equation 2x + 4x² = 9 is 4x² + 2x - 9 = 0.
- The type of solutions indicated by a discriminant of 16 corresponds to unequal, rational roots.
Algebraic Expressions and Simplification
- The imaginary form of √-16 is represented as 4i.
- Simplifying the expression 3i/(2-i) requires multiplying by 2 + i in both the numerator and denominator.
Solving Equations and Systems
- For the equation 3x − 4 = 8, the solution is x = 4.
- The equation ax = 2x + 5 can be solved to find x = (5/a - 2).
- Absolute value equations like 2|x + 2| = 4 yield solutions x = -4, 0.
Finding Roots
- For the quadratic x² − 3x − 10 = 0, the solutions are x = -2, 5.
- The equation x² − 15 = 1 results in solutions x = ±4.
Inequalities
- The inequality -8x < 8 has the solution (-1, ∞).
- The compound inequality 4 ≤ 2x ≤ 10 translates to the interval [2, 5].
Absolute Value and Complex Numbers
- The absolute value inequality |x + 1| > 3 results in two intervals: (-∞, -4) or (2, ∞).
- The complex number i raised to an exponent, i^15, simplifies to -i.
Operations with Complex Numbers
- The result of (2 + 7i) − (4 − 8i) is -2 + 15i.
- The product (6 − 2i)(1 + i) computes to 8 + 4i.
- The fraction 3/(6+i) simplifies to 18/37 - (3i/37).
Additional Algebraic Problems
- For the equation 1/2z + 3z = 5/3z - 11, the solution is z = -6.
- In the equation 3/x - 2 = 8/x + 6, solutions are found at x = 34/5 or x = 6 4/5.
- The equation 2/x = 3/(x+1) + 2/(x+1) simplifies to find x = 2/3.
- Solving √3x + 1 = 4 leads to x = 3.
- The equation √5x + 5 = √4x + 1 gives the solution x = 4.
Fraction Problems
- A fraction where the denominator is 4 more than its numerator, resulting in a value of 1/3, equates to 2/6.
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