Ch. 1 - Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problema 71

Capitolo 2, Problema 71

In Exercises 59–94, solve each absolute value inequality. |x - 1| ≥ 2

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Recall that an absolute value inequality of the form $|A| \geq B$ (where $B > 0$) can be rewritten as two separate inequalities: $A \leq -B$ or $A \geq B$.

Identify the expression inside the absolute value: here, $A = x - 1$ and $B = 2$.

Set up the two inequalities based on the rule: $x - 1 \leq -2$ or $x - 1 \geq 2$.

Solve each inequality separately: For $x - 1 \leq -2$, add 1 to both sides to get $x \leq -1$. For $x - 1 \geq 2$, add 1 to both sides to get $x \geq 3$.

Combine the solutions to write the final answer as $x \leq -1$ or $x \geq 3$.

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Absolute Value Definition

The absolute value of a number represents its distance from zero on the number line, always as a non-negative value. For any expression |A|, it equals A if A is non-negative, and -A if A is negative. Understanding this helps in rewriting absolute value inequalities into equivalent expressions without absolute values.

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Vertex Form

Solving Absolute Value Inequalities

Absolute value inequalities like |x - 1| ≥ 2 split into two cases: either the expression inside is greater than or equal to 2, or less than or equal to -2. This leads to two separate inequalities to solve, reflecting the distance being at least 2 units away from 1 on the number line.

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Linear Inequalities

Number Line Interpretation

Interpreting absolute value inequalities on a number line helps visualize the solution set. For |x - 1| ≥ 2, the solutions are all points at least 2 units away from 1, meaning x ≤ -1 or x ≥ 3. This visualization aids in understanding and verifying the solution intervals.

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The Slope of a Line

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