In Exercises 91–100, find all values of x satisfying the given conditions. y = |2 - 3x| and y = 13

검증된 단계별 안내

Start with the given equations: \(y = |2 - 3x|\) and \(y = 13\). Since both expressions equal \(y\), set them equal to each other: \(|2 - 3x| = 13\).

Recall that the absolute value equation \(|A| = B\) (where \(B > 0\)) splits into two cases: \(A = B\) or \(A = -B\). Apply this to \(|2 - 3x| = 13\) to get two equations: \(2 - 3x = 13\) and \(2 - 3x = -13\).

Solve the first equation \(2 - 3x = 13\) by isolating \(x\): subtract 2 from both sides to get \(-3x = 11\), then divide both sides by \(-3\) to find \(x\).

Solve the second equation \(2 - 3x = -13\) similarly: subtract 2 from both sides to get \(-3x = -15\), then divide both sides by \(-3\) to find \(x\).

The solutions from both cases give all values of \(x\) that satisfy the original equation \(|2 - 3x| = 13\).

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주요 개념

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

Absolute Value Function

The absolute value of a number represents its distance from zero on the number line, always yielding a non-negative result. For an expression like |2 - 3x|, it means the value inside the bars can be positive or negative, but the output is always positive or zero.

Solving Absolute Value Equations

To solve equations involving absolute values, set the expression inside the absolute value equal to both the positive and negative of the given value. For |2 - 3x| = 13, solve 2 - 3x = 13 and 2 - 3x = -13 separately to find all possible x values.

Linear Equations

Linear equations are algebraic expressions of the first degree, meaning variables are not raised to any power other than one. After removing the absolute value, solving 2 - 3x = ±13 involves isolating x using basic algebraic operations like addition, subtraction, multiplication, and division.