Solve each absolute value inequality. - 4|1 - x| \u003c - 16

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| \u003e 4. Step 2: Recall the definition of absolute value inequalities. For |A| \u003e B, the inequality splits into two cases: A \u003e B or A . Apply this to the inequality: 1 - x \u003e 4 or 1 - x . Step 3: Solve each case separately. For the first case, 1 - x \u003e 4, subtract 1 from both sides to isolate -x: -x \u003e 3. Then divide by -1 (reversing the inequality sign): x . Step 4: For the second case, 1 - x , subtract 1 from both sides to isolate -x: -x . Then divide by -1 (reversing the inequality sign): x \u003e 5. Step 5: Combine the solutions from both cases. The solution to the inequality is x or x \u003e 5. This represents the values of x that satisfy the original inequality.

Ch. 1 - Equations and Inequalities

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Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

문제 83a

2장, 문제 83a

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

검증된 단계별 안내

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

A < -B

. Apply this to the inequality:

1 - x > 4

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

x > 5

. This represents the values of

x

that satisfy the original inequality.

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주요 개념

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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.

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.

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.