In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. |x + 1| + 5 = 3

검증된 단계별 안내

Start with the given equation: \(|x + 1| + 5 = 3\).

Isolate the absolute value expression by subtracting 5 from both sides: \(|x + 1| = 3 - 5\).

Simplify the right side: \(|x + 1| = -2\).

Recall that the absolute value of any expression is always greater than or equal to zero, so \(|x + 1| = -2\) has no solution because an absolute value cannot be negative.

Conclude that the equation has no solution.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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| = B, if B is negative, there is no solution because absolute values cannot be negative.

Solving Absolute Value Equations

To solve an equation involving absolute value, isolate the absolute value expression first. Then, set up two separate equations: one where the inside expression equals the positive value, and one where it equals the negative value, solving each separately.

Checking for No Solution

If isolating the absolute value results in a negative number on the right side, the equation has no solution. This is because absolute values cannot be negative, so no real number satisfies the equation.