Questions and Answers
What is the standard form of a linear equation in one variable?
ax + b = 0
Is the equation 6x^2 + 1 = 0 linear in one variable?
False
Is the equation 3/x - 9 = 0 linear in one variable?
False
Is the equation 2x + 4y = 5 linear in one variable?
Signup and view all the answers
Is the equation x/3 - 9 = 0 linear in one variable?
Signup and view all the answers
What is the solution for y in the equation (y + 5)/2 - (y - 2)/4 = (y + 7)/3 + 1?
Signup and view all the answers
What is the value of m in the equation 5/(m - 2) = 3m/(m^2 + 2m - 8) - 2/(m + 4)?
Signup and view all the answers
What is the expression for b when solving the equation A = 1/2h(B + b)?
Signup and view all the answers
Kent invested a total of $8000. How much did he invest in the stock fund and the CD?
Signup and view all the answers
How much 4% acid solution should be mixed with 200 mL of a 12% acid solution to make a 9% acid solution?
Signup and view all the answers
What is the square root of -12?
Signup and view all the answers
What is the value of the square root of -15 times the square root of -3?
Signup and view all the answers
What is the result of square root of -50 divided by square root of -2?
Signup and view all the answers
What is the real part of the complex number 3 + 4i?
Signup and view all the answers
What is the imaginary part of the complex number (1/2) - (2/3)i?
Signup and view all the answers
What is the standard form of the complex number 3 - sqrt(-100)?
Signup and view all the answers
What is the simplified form of (-6 + sqrt(-18))/9?
Signup and view all the answers
What is the value of i raised to the power of -19?
Signup and view all the answers
What is the value of i raised to the power of 23?
Signup and view all the answers
How much 10% solution needs to be added to 40cc of 50% solution to make a 35% solution?
Signup and view all the answers
A total amount of $7000 was added into a stock fund and a CD. How much was added into each account?
Signup and view all the answers
What is the sum of the expression (-2 - 4i) + (5 + 2i) - (3 - 6i)?
Signup and view all the answers
What is the result of (a + bi)(a - bi)?
Signup and view all the answers
What is the result of (8 + 2i)/(3 + 5i)?
Signup and view all the answers
What are the roots of the equation n^2 + 5n = 24?
Signup and view all the answers
What are the solutions for x in the equation x^2 = 64?
Signup and view all the answers
What is the solution for w in the equation (w + 3)^2 = 8?
Signup and view all the answers
In the formula S = n/2(a + d), what is the expression to solve for d?
Signup and view all the answers
How much E5 should be mixed with 5000 gallons of E10 to make an E9 mixture?
Signup and view all the answers
What is the result of (6 + 2i)/(3 - i)?
Signup and view all the answers
For the quadratic expression x^2 + 2/9x + n, what value of n makes it a perfect square and what are its factors?
Signup and view all the answers
What is the solution for z in the equation (3z - 18)^2 + 58 = 40?
Signup and view all the answers
What is the process to make a polynomial a perfect square?
Signup and view all the answers
What is the quadratic formula?
Signup and view all the answers
Use the quadratic formula to solve the equation 5x^2 - 6x + 3.
Signup and view all the answers
What is the discriminant in quadratic equations?
Signup and view all the answers
If the discriminant is negative, how many solutions are there?
Signup and view all the answers
If the discriminant is positive, how many solutions are there?
Signup and view all the answers
If the discriminant is zero, how many solutions are there?
Signup and view all the answers
What is the absolute value of -7?
Signup and view all the answers
What are the solutions for the absolute value of 3?
Signup and view all the answers
What are the solutions for the equation 5|2 - 4t| = 50?
Signup and view all the answers
What are the solutions for the equation |3x - 4| = |2x + 1|?
Signup and view all the answers
If 0 = 0, it means it is an identity and pertains to all real numbers.
Signup and view all the answers
What is the solution for x in the equation 2(x - 4)^(3/4)?
Signup and view all the answers
What is the inequality for m in the equation (m - 4)/2 - (3m + 4)/10 > -3/5m?
Signup and view all the answers
Write the set builder notation and interval notation for y < 2.
Signup and view all the answers
Study Notes
Linear Equations
- Linear equations in one variable have the form ( ax + b = 0 ).
- An example of a non-linear equation: ( 6x^2 + 1 = 0 ).
- Fractions in linear equations can make them non-linear, e.g., ( \frac{3}{x} - 9 = 0 ).
- Multi-variable equations, such as ( 2x + 4y = 5 ), are not linear in one variable.
- Equation ( \frac{x}{3} - 9 = 0 ) is linear in one variable.
Solving Equations
- For ( \frac{y+5}{2} - \frac{y-2}{4} = \frac{y+7}{3} + 1 ), the solution is ( y = -4 ).
- The equation ( \frac{5}{m-2} = \frac{3m}{m^2 + 2m - 8} - \frac{2}{m + 4} ) results in ( m = -4 ) with specific restrictions.
Areas and Investments
- The formula for the area of a trapezoid is ( A = \frac{1}{2}h(B + b) ); solving for ( b ) gives ( b = \frac{2A}{h} - B ).
- Investment scenario: Kent invested $6000 in a stock fund at 6.5% for 2 years and $2000 in a CD at 2.5%, yielding total interest of $855.
Acid Solution Mixtures
- To create a 9% acid solution from 200 mL of a 12% solution, mix with 120 mL of a 4% acid solution.
- To produce a 35% solution from 40cc of a 50% solution, add 24cc of a 10% solution.
Complex Numbers
- The square root of (-12) is ( 2i\sqrt{3} ).
- ( \sqrt{-15} \times \sqrt{-3 = -3\sqrt{5} ).
- The value of ( \sqrt{-50} / \sqrt{-2} = 5 ).
- The real part of ( 3 + 4i ) is 3, and the imaginary part of ( \frac{1}{2} - \frac{2}{3}i ) is (-\frac{2}{3}).
- Standard form for ( 3 - \sqrt{-100} ) is ( 3 - 10i ).
Solving Quadratic Equations
- Solutions for ( n^2 + 5n = 24 ) are (-8) and (3).
- Using the square root method, ( x^2 = 64 ) gives ( { \pm 8 } ).
- Completing the square for ( (w+3)^2 = 8 ) yields ( {-3 \pm 2\sqrt{2}} ).
- The quadratic formula is given by ( X = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).
- Discriminant ( b^2 - 4ac ) determines the nature of solutions: negative for non-real solutions, positive for two real solutions, and zero for one real solution.
Absolute Values and Inequalities
- The absolute value of (-7) is (7), and for (3), it is {-3, 3}.
- The equation ( 5|2 - 4t| = 50 ) results in two solutions: {-2, 3}.
- The equation ( |3x - 4| = |2x + 1| ) has solutions ( {5, \frac{3}{5}} ).
Set Notation and Intervals
- An identity equation ( 0 = 0 ) implies it is valid for all real numbers.
- For ( 2(x - 4)^{3/4} ), solving for ( x ) results in ( x = 85 ).
- The inequality ( \frac{m - 4}{2} - \frac{3m + 4}{10} > -\frac{3}{5}m ) simplifies to ( m > 3 ), with set-builder notation as ( {m | m > 3} ) and interval notation as ( (3, \infty) ).
- For ( y_2 ), set-builder notation is ( {y | y < -4} ) and interval notation is ( (-\infty, -4) \cup (2, \infty) ).
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your knowledge on linear equations in one variable with this set of flashcards. Perfect for reviewing the key concepts and differentiating between linear and non-linear equations. Challenge yourself and enhance your understanding of algebraic principles.