Topics in Algebra: Strategies for Solving Equations

Learn to isolate the variable using inverse operations, focusing on addition and subtraction to solve equations. Check solutions by substitution and apply the method to multiple examples.

Isolate the variable using inverse operations with multiplication and division. Practice solving equations, including fractions and reciprocal strategies, and verify results with substitution.

Learn a five-step approach to solving multi-step equations, including distribution, combining like terms, isolating the variable, and applying multiplication or division with substitution checks.

Explore how special cases arise in solving equations: contradictions yield no solution, identities yield all real numbers, with distribution and variable elimination illustrated.

Use a common denominator to clear fractions, selecting denominators such as 30 or 12, multiply through, perform cancellations, and solve for x, embracing fractional answers when needed.

This lecture demonstrates clearing decimals by multiplying through with a power of ten, then applying a five-step process to solve equations and isolate x.

Learn to rewrite formulas and literal equations to isolate the indicated variable using inverse operations, including using reciprocals and division to solve for h, g, and w.

Graph inequalities on the number line and express solutions in interval notation, using open or closed boundaries and ranges for x-values like greater than 4 or between -2 and 3.

Apply the five-step method for solving equations to linear inequalities, using inverse operations and inequality symbols, and express results on a number line or in interval notation.

Learn how multiplying or dividing by a negative number reverses the inequality symbol in linear inequalities, with examples using number lines and reciprocals to solve and check solutions.

Identify two special inequality cases: a true statement after cancelling variables yields all real numbers, and a false statement yields no solution, with interval or set notation explained.

Identify identities that hold for all real numbers and express the solution as all real numbers, then recognize contradictions that yield no solution or the empty set.

Apply a common denominator strategy to clear fractions in rational equations, multiply through by the LCD, and simplify to solve for x using addition, subtraction, and division.

isolate the radical, square both sides, then solve the resulting equation and verify with substitution, noting domain restrictions and possible extraneous solutions.

Isolate radicals, raise both sides to the appropriate power, and use substitution to check extraneous solutions when solving radical equations with two radicals.

Apply the square root property to solve quadratic equations by isolating the squared term and taking square roots, yielding both positive and negative solutions.

Complete the square to form a perfect square trinomial and use the square root property to solve quadratics. Isolate x-terms, add the constant, and solve x with plus or minus.

Explore how to recognize quadratic forms beyond x^2, use substitution with u to replace x^2, and solve by factoring, zero product rule, and back-substitution to obtain all solutions.

discover how to solve quadratic equations with the quadratic formula, using coefficients a, b, and c, computing the discriminant, and extracting real or imaginary solutions.

Explore how the discriminant, under the quadratic formula's square root, determines solution count and type: positive gives two real, negative gives two imaginary, zero gives one real (multiplicity two).

Explore compound inequalities that require two conditions simultaneously and learn to solve them by splitting into two linear inequalities, then represent the intersection as interval notation on a number line.

Isolate the absolute values and solve for both the positive and negative results, as shown by examples yielding x = 1 or -4, and x = 8/3 or -2.

Split absolute value equations on both sides into two cases, solve each, and identify no-solution scenarios when the absolute value equals a negative.

Isolate the absolute values, form two inequalities, and solve by flipping the sign when multiplying by negative; then express the solution as interval notation on the number line.