Algebra: Expressions, Linear Equations, Inequalities, and Word Problems
Core skills
Translate words into symbols: sum, more than, and increased by indicate addition; difference, less than, and decreased by indicate subtraction, with less than reversing the order (5 less than x is x - 5); product and of indicate multiplication; quotient, per, and divided by indicate division. Combine like terms only when the variable part is identical, and distribute a factor across every term inside parentheses, including the sign in front. Solve linear equations by undoing operations in reverse order, performing the same operation on both sides, and clearing fractions by multiplying every term by the common denominator. If a variable appears on both sides, collect it on one side first. Inequalities work the same way except that multiplying or dividing both sides by a negative number flips the inequality symbol. For word problems, assign a variable to the unknown, write an equation from the stated relationship, solve, then reread the question, because many items ask for a related quantity rather than the variable itself.
Traps
Dropping a minus sign when distributing a negative; forgetting to flip the inequality; dividing only one side of an equation by a coefficient; solving correctly but answering with the wrong quantity; and mishandling consecutive-integer setups (n, n + 1, n + 2).
Worked example
Solve 5(x - 3) + 2 = 3x + 7. Distribute: 5x - 15 + 2 = 3x + 7, so 5x - 13 = 3x + 7. Subtract 3x: 2x - 13 = 7. Add 13: 2x = 20, so x = 10. Substitute back into the original equation to check.