Linear Equations and Inequalities
Linear Equations and Inequalities
Questions here ask you to solve multi-step equations and inequalities, work with slope and the three standard forms of a line, decide whether lines are parallel or perpendicular, and translate real contexts (rate plus fixed fee, break-even points, ticket sales) into a linear model.
Core rules
- Slope between two points: m = (y₂ − y₁)/(x₂ − x₁). Slope-intercept: y = mx + b. Point-slope: y − y₁ = m(x − x₁). Standard: Ax + By = C.
- Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals, so m₁·m₂ = −1.
- Undo operations in reverse order. Multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol.
- Absolute value: |x| = a gives x = a or x = −a; |x| < a gives −a < x < a (a conjunction); |x| > a gives x < −a or x > a (a disjunction).
- A horizontal line y = b has slope 0; a vertical line x = a has undefined slope and is not a function of x.
Worked example
Write the equation of the line through (−2, 5) that is perpendicular to y = (2/3)x − 4.
The given slope is 2/3, so the perpendicular slope is the negative reciprocal, −3/2. Point-slope gives y − 5 = −(3/2)(x + 2). Distributing: y − 5 = −(3/2)x − 3, so y = −(3/2)x + 2. Check by substituting x = −2: y = 3 + 2 = 5. ✓
Traps that cost marks
- Forgetting to flip ≤ to ≥ after dividing by a negative.
- Using the reciprocal 3/2 instead of the negative reciprocal −3/2.
- Subtracting coordinates in inconsistent order in the slope formula - keep the points in the same order in numerator and denominator.
- Writing a vertical line in the form y = a.
- Accepting an absolute-value solution without checking it in the original equation.