Solve \left|4x+2\right|+\left|2x-5\right|=11

Evaluate

∣ 4 x + 2 ∣ + ∣ 2 x - 5 ∣ = 11

Move the expression to the left side

∣ 4 x + 2 ∣ + ∣ 2 x - 5 ∣ - 11 = 0

Separate the equation into 4 possible cases

4 x + 2 + 2 x - 5 - 11 = 0, 4 x + 2 \geq 0, 2 x - 5 \geq 0 4 x + 2 - (2 x - 5) - 11 = 0, 4 x + 2 \geq 0, 2 x - 5 < 0 - (4 x + 2) + 2 x - 5 - 11 = 0, 4 x + 2 < 0, 2 x - 5 \geq 0 - (4 x + 2) - (2 x - 5) - 11 = 0, 4 x + 2 < 0, 2 x - 5 < 0

Solve the equation

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Evaluate

4 x + 2 + 2 x - 5 - 11 = 0

Calculate the sum or difference

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Evaluate

4 x + 2 + 2 x - 5 - 11

Add the terms

6 x + 2 - 5 - 11

Subtract the numbers

6 x - 14

6 x - 14 = 0

Move the constant to the right-hand side and change its sign

6 x = 0 + 14

Removing 0 doesn't change the value,so remove it from the expression

6 x = 14

Divide both sides

\frac{6 x}{6} = \frac{14}{6}

Divide the numbers

x = \frac{14}{6}

Cancel out the common factor 2

x = \frac{7}{3}

x = \frac{7}{3}, 4 x + 2 \geq 0, 2 x - 5 \geq 0 4 x + 2 - (2 x - 5) - 11 = 0, 4 x + 2 \geq 0, 2 x - 5 < 0 - (4 x + 2) + 2 x - 5 - 11 = 0, 4 x + 2 < 0, 2 x - 5 \geq 0 - (4 x + 2) - (2 x - 5) - 11 = 0, 4 x + 2 < 0, 2 x - 5 < 0

Solve the inequality

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Evaluate

4 x + 2 \geq 0

Move the constant to the right side

4 x \geq 0 - 2

Removing 0 doesn't change the value,so remove it from the expression

4 x \geq - 2

Divide both sides

\frac{4 x}{4} \geq \frac{- 2}{4}

Divide the numbers

x \geq \frac{- 2}{4}

Divide the numbers

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Evaluate

\frac{- 2}{4}

Cancel out the common factor 2

\frac{- 1}{2}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

- \frac{1}{2}

x \geq - \frac{1}{2}

x = \frac{7}{3}, x \geq - \frac{1}{2}, 2 x - 5 \geq 0 4 x + 2 - (2 x - 5) - 11 = 0, 4 x + 2 \geq 0, 2 x - 5 < 0 - (4 x + 2) + 2 x - 5 - 11 = 0, 4 x + 2 < 0, 2 x - 5 \geq 0 - (4 x + 2) - (2 x - 5) - 11 = 0, 4 x + 2 < 0, 2 x - 5 < 0

Solve the inequality

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Evaluate

2 x - 5 \geq 0

Move the constant to the right side

2 x \geq 0 + 5

Removing 0 doesn't change the value,so remove it from the expression

2 x \geq 5

Divide both sides

\frac{2 x}{2} \geq \frac{5}{2}

Divide the numbers

x \geq \frac{5}{2}

x = \frac{7}{3}, x \geq - \frac{1}{2}, x \geq \frac{5}{2} 4 x + 2 - (2 x - 5) - 11 = 0, 4 x + 2 \geq 0, 2 x - 5 < 0 - (4 x + 2) + 2 x - 5 - 11 = 0, 4 x + 2 < 0, 2 x - 5 \geq 0 - (4 x + 2) - (2 x - 5) - 11 = 0, 4 x + 2 < 0, 2 x - 5 < 0

Solve the equation

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Evaluate

4 x + 2 - (2 x - 5) - 11 = 0

Calculate

4 x + 2 - 2 x + 5 - 11 = 0

Calculate the sum or difference

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Evaluate

4 x + 2 - 2 x + 5 - 11

Subtract the terms

2 x + 2 + 5 - 11

Calculate the sum or difference

2 x - 4

2 x - 4 = 0

Move the constant to the right-hand side and change its sign

2 x = 0 + 4

Removing 0 doesn't change the value,so remove it from the expression

2 x = 4

Divide both sides

\frac{2 x}{2} = \frac{4}{2}

Divide the numbers

x = \frac{4}{2}

Divide the numbers

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Evaluate

\frac{4}{2}

Reduce the numbers

\frac{2}{1}

Calculate

2

x = 2

x = \frac{7}{3}, x \geq - \frac{1}{2}, x \geq \frac{5}{2} x = 2, 4 x + 2 \geq 0, 2 x - 5 < 0 - (4 x + 2) + 2 x - 5 - 11 = 0, 4 x + 2 < 0, 2 x - 5 \geq 0 - (4 x + 2) - (2 x - 5) - 11 = 0, 4 x + 2 < 0, 2 x - 5 < 0

Solve the inequality

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Evaluate

4 x + 2 \geq 0

Move the constant to the right side

4 x \geq 0 - 2

Removing 0 doesn't change the value,so remove it from the expression

4 x \geq - 2

Divide both sides

\frac{4 x}{4} \geq \frac{- 2}{4}

Divide the numbers

x \geq \frac{- 2}{4}

Divide the numbers

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Evaluate

\frac{- 2}{4}

Cancel out the common factor 2

\frac{- 1}{2}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

- \frac{1}{2}

x \geq - \frac{1}{2}

x = \frac{7}{3}, x \geq - \frac{1}{2}, x \geq \frac{5}{2} x = 2, x \geq - \frac{1}{2}, 2 x - 5 < 0 - (4 x + 2) + 2 x - 5 - 11 = 0, 4 x + 2 < 0, 2 x - 5 \geq 0 - (4 x + 2) - (2 x - 5) - 11 = 0, 4 x + 2 < 0, 2 x - 5 < 0

Solve the inequality

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Evaluate

2 x - 5 < 0

Move the constant to the right side

2 x < 0 + 5

Removing 0 doesn't change the value,so remove it from the expression

2 x < 5

Divide both sides

\frac{2 x}{2} < \frac{5}{2}

Divide the numbers

x < \frac{5}{2}

x = \frac{7}{3}, x \geq - \frac{1}{2}, x \geq \frac{5}{2} x = 2, x \geq - \frac{1}{2}, x < \frac{5}{2} - (4 x + 2) + 2 x - 5 - 11 = 0, 4 x + 2 < 0, 2 x - 5 \geq 0 - (4 x + 2) - (2 x - 5) - 11 = 0, 4 x + 2 < 0, 2 x - 5 < 0

Solve the equation

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Evaluate

- (4 x + 2) + 2 x - 5 - 11 = 0

Calculate

- 4 x - 2 + 2 x - 5 - 11 = 0

Calculate the sum or difference

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Evaluate

- 4 x - 2 + 2 x - 5 - 11

Add the terms

- 2 x - 2 - 5 - 11

Subtract the numbers

- 2 x - 18

- 2 x - 18 = 0

Move the constant to the right-hand side and change its sign

- 2 x = 0 + 18

Removing 0 doesn't change the value,so remove it from the expression

- 2 x = 18

Change the signs on both sides of the equation

2 x = - 18

Divide both sides

\frac{2 x}{2} = \frac{- 18}{2}

Divide the numbers

x = \frac{- 18}{2}

Divide the numbers

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Evaluate

\frac{- 18}{2}

Reduce the numbers

\frac{- 9}{1}

Calculate

- 9

x = - 9

x = \frac{7}{3}, x \geq - \frac{1}{2}, x \geq \frac{5}{2} x = 2, x \geq - \frac{1}{2}, x < \frac{5}{2} x = - 9, 4 x + 2 < 0, 2 x - 5 \geq 0 - (4 x + 2) - (2 x - 5) - 11 = 0, 4 x + 2 < 0, 2 x - 5 < 0

Solve the inequality

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Evaluate

4 x + 2 < 0

Move the constant to the right side

4 x < 0 - 2

Removing 0 doesn't change the value,so remove it from the expression

4 x < - 2

Divide both sides

\frac{4 x}{4} < \frac{- 2}{4}

Divide the numbers

x < \frac{- 2}{4}

Divide the numbers

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Evaluate

\frac{- 2}{4}

Cancel out the common factor 2

\frac{- 1}{2}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

- \frac{1}{2}

x < - \frac{1}{2}

x = \frac{7}{3}, x \geq - \frac{1}{2}, x \geq \frac{5}{2} x = 2, x \geq - \frac{1}{2}, x < \frac{5}{2} x = - 9, x < - \frac{1}{2}, 2 x - 5 \geq 0 - (4 x + 2) - (2 x - 5) - 11 = 0, 4 x + 2 < 0, 2 x - 5 < 0

Solve the inequality

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Evaluate

2 x - 5 \geq 0

Move the constant to the right side

2 x \geq 0 + 5

Removing 0 doesn't change the value,so remove it from the expression

2 x \geq 5

Divide both sides

\frac{2 x}{2} \geq \frac{5}{2}

Divide the numbers

x \geq \frac{5}{2}

x = \frac{7}{3}, x \geq - \frac{1}{2}, x \geq \frac{5}{2} x = 2, x \geq - \frac{1}{2}, x < \frac{5}{2} x = - 9, x < - \frac{1}{2}, x \geq \frac{5}{2} - (4 x + 2) - (2 x - 5) - 11 = 0, 4 x + 2 < 0, 2 x - 5 < 0

Solve the equation

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Evaluate

- (4 x + 2) - (2 x - 5) - 11 = 0

Calculate

- 4 x - 2 - 2 x + 5 - 11 = 0

Calculate the sum or difference

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Evaluate

- 4 x - 2 - 2 x + 5 - 11

Subtract the terms

- 6 x - 2 + 5 - 11

Calculate the sum or difference

- 6 x - 8

- 6 x - 8 = 0

Move the constant to the right-hand side and change its sign

- 6 x = 0 + 8

Removing 0 doesn't change the value,so remove it from the expression

- 6 x = 8

Change the signs on both sides of the equation

6 x = - 8

Divide both sides

\frac{6 x}{6} = \frac{- 8}{6}

Divide the numbers

x = \frac{- 8}{6}

Divide the numbers

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Evaluate

\frac{- 8}{6}

Cancel out the common factor 2

\frac{- 4}{3}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

- \frac{4}{3}

x = - \frac{4}{3}

x = \frac{7}{3}, x \geq - \frac{1}{2}, x \geq \frac{5}{2} x = 2, x \geq - \frac{1}{2}, x < \frac{5}{2} x = - 9, x < - \frac{1}{2}, x \geq \frac{5}{2} x = - \frac{4}{3}, 4 x + 2 < 0, 2 x - 5 < 0

Solve the inequality

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Evaluate

4 x + 2 < 0

Move the constant to the right side

4 x < 0 - 2

Removing 0 doesn't change the value,so remove it from the expression

4 x < - 2

Divide both sides

\frac{4 x}{4} < \frac{- 2}{4}

Divide the numbers

x < \frac{- 2}{4}

Divide the numbers

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Evaluate

\frac{- 2}{4}

Cancel out the common factor 2

\frac{- 1}{2}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

- \frac{1}{2}

x < - \frac{1}{2}

x = \frac{7}{3}, x \geq - \frac{1}{2}, x \geq \frac{5}{2} x = 2, x \geq - \frac{1}{2}, x < \frac{5}{2} x = - 9, x < - \frac{1}{2}, x \geq \frac{5}{2} x = - \frac{4}{3}, x < - \frac{1}{2}, 2 x - 5 < 0

Solve the inequality

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Evaluate

2 x - 5 < 0

Move the constant to the right side

2 x < 0 + 5

Removing 0 doesn't change the value,so remove it from the expression

2 x < 5

Divide both sides

\frac{2 x}{2} < \frac{5}{2}

Divide the numbers

x < \frac{5}{2}

x = \frac{7}{3}, x \geq - \frac{1}{2}, x \geq \frac{5}{2} x = 2, x \geq - \frac{1}{2}, x < \frac{5}{2} x = - 9, x < - \frac{1}{2}, x \geq \frac{5}{2} x = - \frac{4}{3}, x < - \frac{1}{2}, x < \frac{5}{2}

Find the intersection

x \in \emptyset x = 2, x \geq - \frac{1}{2}, x < \frac{5}{2} x = - 9, x < - \frac{1}{2}, x \geq \frac{5}{2} x = - \frac{4}{3}, x < - \frac{1}{2}, x < \frac{5}{2}

Find the intersection

x \in \emptyset x = 2 x = - 9, x < - \frac{1}{2}, x \geq \frac{5}{2} x = - \frac{4}{3}, x < - \frac{1}{2}, x < \frac{5}{2}

Find the intersection

x \in \emptyset x = 2 x \in \emptyset x = - \frac{4}{3}, x < - \frac{1}{2}, x < \frac{5}{2}

Find the intersection

x \in \emptyset x = 2 x \in \emptyset x = - \frac{4}{3}

Find the union

x = 2 x = - \frac{4}{3}

Solution

x_{1} = - \frac{4}{3}, x_{2} = 2

Alternative Form

x_{1} = - 1. \dot{3}, x_{2} = 2

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