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College Algebra Unit 3: Linear equations & inequalities

Linear equations and inequalities are solved by undoing operations in reverse order. Absolute value adds a twist: it creates two cases.

Covers: Solving linear eqns, absolute value eqns/inequalities, compound inequalities, modeling

Study guide

Linear equations are solved with inverse operations: clear fractions by multiplying by the LCD, distribute, combine like terms, collect the variable on one side, and divide. An equation can have one solution, no solution (a false statement like 0 = 5), or infinitely many (an identity like 0 = 0).

Absolute value measures distance from zero. The equation |x| = a with a > 0 means x = a or x = −a, so isolate the absolute value first and split into two equations. If a r means x c + r.

Linear inequalities are solved like equations, but multiplying or dividing by a negative reverses the inequality. Compound inequalities joined by "and" (like −3 < 2x + 1 ≤ 7) are solved by doing the same operation to all three parts; "or" inequalities are solved separately and combined.

Write solutions in interval notation: brackets include endpoints and parentheses exclude them, so x ≥ 2 is [2, ∞) and −1 < x < 5 is (−1, 5). Modeling problems translate a situation into an equation or inequality, such as total cost = fixed cost + rate × quantity.

The slope through (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁)/(x₂ − x₁). A line can be written y = mx + b (slope-intercept form), y − y₁ = m(x − x₁) (point-slope form), or Ax + By = C. Find intercepts by setting x = 0 or y = 0. Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals, so m₁m₂ = −1. A horizontal line y = k has slope 0, and a vertical line x = k has undefined slope.

y varies directly with x if y = kx, inversely if y = k/x, and inversely with the square if y = k/x². Find k from one known pair of values, then use the equation to predict others. An equation |A| = |B| means A = B or A = −B. Solve both equations and keep every solution.

Key ideas

• Do the same operation to both sides to isolate the variable.
• Multiplying or dividing an inequality by a negative flips the inequality sign.
• |x| = a (a > 0) means x = a or x = −a.
• |x| a becomes x a (an "or").
• Write solution sets in interval notation: parentheses for open ends, brackets for included ends.

Formulas and rules

Common mistakes

• Forgetting to flip the inequality when dividing by a negative.
• Solving |x − 3| = −2 instead of noticing it has no solution.

Words to know

Inequality sign reversal

Multiplying or dividing by a negative flips the sign.

Absolute value equation

|x| = a (a > 0) gives x = a or x = −a.

Absolute value inequality

"Less than" is an and-statement: |x − c| < r → c − r < x < c + r.

Absolute value

Distance from 0; never negative, so |x| = −4 has no solution.

Interval notation

Brackets include endpoints, parentheses don't: x ≥ 2 is [2, ∞).

Slope-intercept form

y = mx + b

Linear equation

ax + b = 0

Identity

An equation true for every value, giving 0 = 0.

Contradiction (equation)

An equation with no solution, giving a false statement.

Compound inequality

Two inequalities joined by "and" or "or."

"Greater than" absolute value

|x − c| > r means x c + r.

Linear model

Total = fixed amount + rate × quantity.

Solving a linear equation

Clear fractions, distribute, combine like terms, collect the variable, divide.

Sample practice questions

33 more questions like these, from level 1 to 5, are in the practice set.