Equations and Inequalities

This lecture teaches a five-step blueprint for solving degree one equations: remove fractions, remove parentheses, move and combine like terms, simplify, then isolate the variable by dividing its coefficient.

Apply the five-step method to solve a degree 1 equation: remove fractions, remove parentheses, move like terms, simplify, and divide by the coefficient.

Follow five basic rules to solve a degree one equation: remove fractions and parentheses, move like terms, simplify, and finally divide by the coefficient to isolate x.

Explore the form of second degree equations, ax^2 + bx + c, where a is not zero, and the quadratic formula to solve.

Explore the quadratic formula for solving second-degree equations, using the discriminant to determine two distinct solutions, a repeated solution, or no solution, with applications in engineering, physics, and chemistry.

Apply the quadratic formula to solve ax^2+bx+c=0, compute the discriminant, take its square root, and find two solutions x1 and x2, such as -1/2 and -3.

Learn to convert a quadratic to standard form, identify a, b, c, and apply the quadratic formula to solve degree-2 equations, including handling the discriminant and preserving exact fractions.

Solve a degree two equation using the quadratic formula by converting to standard form; with a=2, b=0, c=10, the discriminant is negative, so the equation has no solution.

Solve a degree-2 equation using the quadratic formula, verify standard form, identify a, b, and c, and obtain a single repeated root when the discriminant is zero.

Learn to express one variable in terms of another by isolating it in a two-variable equation, yielding a = (5 - 3b)/2.

Define the system of equations as a group of equations with multiple variables, illustrated by two equations in x and y, and prepare to learn methods to solve them.

apply the substitution method to solve a two-variable system by rewriting one equation in terms of a variable, substituting into the other, and solving for a and b.

Apply the substitution method to solve a two-variable system by making x the subject, substituting into the second equation, and solving for y to determine x.

Learn the elimination method for solving systems of linear equations with two variables by adding equations to cancel a variable, solving for one variable, and back-substituting to find the other.

solve a two-equation system using the elimination method by canceling a variable, multiply the second equation by -2, find y, then x, and verify with substitution.

Explore algebraic inequalities by defining left and right sides, reading 2x < 3y + 4, and distinguishing less-than and greater-than relationships from equations.

Subtract the same number from both sides to keep the left and right sides balanced. Apply the subtraction property to decimals, fractions, and other numbers while preserving the inequality sign.

Explore the multiplication property of algebraic inequalities, showing how to multiply both sides by a positive number, and how to handle negatives by flipping the inequality, with examples.

Divide algebraic inequalities by the same number, using positive divisors without flipping, while negative numbers require flipping the inequality; examples include c > 3 and d > -7/3.

Apply the squaring operation to both sides of an inequality to keep it unaffected, with square roots canceling against squares.

Apply the square root to both sides of a positive inequality, per property 6 of algebraic inequalities. Then h^2 < 36 becomes h < 6.

Apply the reciprocal to sides of an inequality, then flip the inequality sign; for example, 1/c > 10 gives c < 1/10, and 1/b < 1/59 gives b > 59.

Practice solving algebraic inequalities by clearing denominators and flipping the inequality when multiplying by a negative, yielding the solution set a < -7.5.