WEBVTT

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Here is the problem: |2x plus 5| greater than 6.

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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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Split into two ("or") inequalities. The inside must be at least c units away from zero, on either side.

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Find the boundary points. The solution lies outside the endpoints.

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Write the solution set in interval notation.

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So the answer is x less than negative 11 over 2 or x greater than 1 over 2; interval notation: (−∞, negative 11 over 2) ∪ (1 over 2, ∞).

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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 5 over 2 (between the endpoints): |0| equals 0, which is not greater than 6, so the middle region is excluded.