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Ch. 1 - Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 1 - Equations and InequalitiesProblema 83a
Capitolo 2, Problema 83a

Solve each absolute value inequality. - 4|1 - x| < - 16

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Step 1: Start by isolating the absolute value expression. Divide both sides of the inequality by -4. Remember, dividing by a negative number reverses the inequality sign. The inequality becomes: |1 - x| > 4.
Step 2: Recall the definition of absolute value inequalities. For |A| > B, the inequality splits into two cases: A > B or A < -B. Apply this to the inequality: 1 - x > 4 or 1 - x < -4.
Step 3: Solve each case separately. For the first case, 1 - x > 4, subtract 1 from both sides to isolate -x: -x > 3. Then divide by -1 (reversing the inequality sign): x < -3.
Step 4: For the second case, 1 - x < -4, subtract 1 from both sides to isolate -x: -x < -5. Then divide by -1 (reversing the inequality sign): x > 5.
Step 5: Combine the solutions from both cases. The solution to the inequality is x < -3 or x > 5. This represents the values of x that satisfy the original inequality.

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Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. It is denoted by vertical bars, such as |x|, and is always non-negative. For example, |3| = 3 and |-3| = 3. Understanding absolute value is crucial for solving inequalities that involve expressions within these bars.
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Inequalities

Inequalities express a relationship where one side is not equal to the other, using symbols like <, >, ≤, or ≥. In the context of absolute value inequalities, they indicate the range of values that satisfy the condition. For instance, solving |x| < a means finding all x values that are within a distance a from zero.
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Properties of Inequalities

When manipulating inequalities, certain properties must be observed, such as the fact that multiplying or dividing by a negative number reverses the inequality sign. This is essential when isolating variables in absolute value inequalities. Understanding these properties helps ensure that the solutions derived from the inequalities are valid.
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The equations in Exercises 79–90 combine the types of equations we have discussed in this section. Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 2/x + 1/2 = 3/4

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