Determine whether each value of x is a solution of the inequality. Inequality. x^2-x-12 ≥0 (a)

step 1 If we had the inequality, x squared minus x and minus 12, is greater than equal to zero, we want

Determine whether each value of x is a solution of the...

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Key Concepts

Substitution for Verification

Substitution involves directly replacing the variable with specific numerical values to check if the inequality holds. This technique serves as a practical method for verifying whether individual candidate solutions satisfy the inequality.

Critical Points and Interval Testing

The zeros of a quadratic function, found through factorization or other methods, divide the number line into distinct intervals. By selecting test points from each interval and evaluating the quadratic expression, one can determine the sign of the expression in that interval and subsequently decide whether the inequality is satisfied.

Quadratic Inequalities

A quadratic inequality is an inequality that involves a quadratic expression, generally in the form ax² + bx + c compared to zero using inequality signs (?, ?, >, <). Understanding quadratic inequalities involves determining the range of variable values that satisfy the inequality, often by finding the roots of the corresponding quadratic equation and analyzing the intervals between them.

Factorization

Factorization is the process of rewriting a quadratic expression as a product of two linear factors. This approach simplifies identifying the critical points—values where the expression equals zero—which are pivotal in determining the intervals over which the inequality holds.

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