Inequalities - I - Magoosh ACT

Algebra

Inequalities - I

Lesson by Mike McGarry

Magoosh Expert

Lesson by Mike McGarry

Magoosh Expert

Summary

Inequalities test whether one quantity is less than, greater than, equal to within a range, or simply not equal to another, and ACT success depends on reading these relationships precisely across all real numbers. Solving inequalities follows many of the same algebra rules as equations, but multiplying or dividing by a negative number reverses the inequality sign.

The ACT uses five key inequality symbols: less than, greater than, less than or equal to, greater than or equal to, and not equal to, and inverse relationships let you reverse the order of expressions when you flip the symbol appropriately.
A variable such as x represents any number unless a restriction is stated, including negatives, zero, fractions, and decimals, so statements like x < 5 and x ≤ 4 are not equivalent because infinitely many numbers lie between 4 and 5.
Addition and subtraction preserve inequalities exactly as they do with equations, so isolating a variable often begins by moving terms from one side to the other.
Multiplication and division require extra care: using a positive number keeps the inequality direction the same, while using a negative number flips the sign, which is essential when solving expressions such as -x > 3.
Inequality solutions can be written algebraically or shown on a number line, where open circles exclude endpoints, solid dots include endpoints, and shaded regions show all valid values.
Compound and rational inequalities require combining restrictions carefully, such as excluding zero from a denominator, recognizing when quantities must be positive, cross-multiplying only when the sign conditions are clear, and flipping both inequality signs when dividing a chained inequality by a negative.

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