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

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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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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.

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Function Composition

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.

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Solving Logarithmic Equations

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.

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