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1.6 Compound and Absolute Value Inequalities Compound inequalities are just more than one inequality at the same time. Sometimes, they are connected by AND. Sometimes, they are connected by OR.

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Solving an “AND” inequality AND means “these two items must both be true”. It is the intersection of two inequalities. Our strategy will be to solve the two inequalities separately and then put the solutions together.

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Solving an “AND” inequality Solve: 11 < 2x + 5 < 19 First, let’s find the two inequalities. To find them, we just include: 1)The middle expression 2)An inequality symbol 3)Everything across the symbol from the middle.

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Solving an “AND” inequality Solve: 11 < 2x + 5 < 19 11 < 2x + 52x + 5 < 19 6 < 2x 3 < x 2x < 14 x < 7 Now it’s time to put our inequalities back together! 3 < x < 7

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Graphing an “AND” inequality Solve: 11 < 2x + 5 < 19 Solution: 3 < x < 7 3 4 5 6 7 3 < x x < 7 3 < x < 7

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Solving an “OR” inequality OR means “at least one of the two items must both be true”. It is the union of two inequalities. Our strategy will be to solve the two inequalities separately and then put the solutions together.

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Solving an “OR” inequality Solve: x - 2 > -3 or x + 4 < -3 Finding the two inequalities is a bit easier; they are already listed separately.

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Solving an “OR” inequality Solve: x - 2 > -3 or x + 4 < -3 x - 2 > -3x + 4 < -3 x > -1x < -7 Now it’s time to put our inequalities back together! x > -1 or x < -7

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Graphing an “OR” inequality Solve: x - 2 > -3 or x + 4 < -3 Solution: x > -1 or x < -7 x > -1 x < -7 x > -1 or x < -7 -7 -6 -5 -4 -3 -2 -1

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Graphing an Absolute Inequality Graph: |x| < 4 This means the distance from zero is < 4 |x| < 4 -6 -4 -2 0 2 4 6 Graph: |x| > 4 |x| > 4 -6 -4 -2 0 2 4 6 This means the distance from zero is > 4

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Solving Absolute Value Inequalities. When we need to use two cases to solve an absolute value problem, treat the problem like an OR inequality for graphing.

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