WEBVTT

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Here is the problem: |4x plus 1| less than 2.

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Solve an absolute value inequality by turning it into a compound inequality and finding the boundary points.

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Write the problem with the absolute value isolated. The absolute value is already alone on one side, so it can be split into cases.

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Rewrite as a compound ("and") inequality. The inside must lie within c units of zero.

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Find the endpoints by solving each boundary equation. The solution lies between the endpoints.

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Write the solution set in interval notation.

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So the answer is negative 3 over 4 less than x less than 1 over 4; interval notation: (negative 3 over 4, 1 over 4).

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An absolute value inequality compares a distance with c: "less than" keeps the inside within c of zero (one interval), "greater than" pushes it beyond c (two rays).

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Test point x equals negative 1 over 4 (midpoint): |0| equals 0 less than 2, so the inside of the interval satisfies the inequality.