Solve the inequalities in Exercises 35-42 . Express the...

Solve the inequalities in Exercises $35-42 .$ Express the solution sets as intervals or unions of intervals and show them on the real line. Use the result $\sqrt{a^{2}}=|a|$ as appropriate.
$$
x^{2}-x<0
$$

Key Concepts

Quadratic Inequalities

Quadratic inequalities involve expressions of the form ax² + bx + c < 0 (or ? 0, > 0, ? 0) and require determining the intervals on the number line where the quadratic expression is either positive or negative. Solving these inequalities usually involves finding the roots of the corresponding quadratic equation, which serve as the boundary points separating different sign regions.

Factoring

Factoring is the process of rewriting a quadratic expression as a product of linear factors. This simplification is crucial in solving quadratic equations and inequalities because it directly reveals the zeros of the function, which are used to partition the number line into intervals where the expression maintains consistent signs.

Sign Analysis

Sign analysis, or the sign chart method, involves testing the sign of each factor in the intervals determined by the zeros of the quadratic. By analyzing whether each factor is positive or negative in these intervals, one can decide where the product of the factors is less than or greater than zero, thereby solving the inequality.

Absolute Value and Square Roots

The property ?(a²) = |a| is a fundamental concept in algebra, particularly when dealing with expressions involving squares and square roots. It ensures that the result of the square root operation is nonnegative and is sometimes employed to simplify or reframe an expression during the process of solving quadratic equations or inequalities.

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