Solve equations with absolute value; including examples and questions with detailed solutions and explanations.

1) | x | = 0 if x = 0

2) | x | = x if x > 0

3) | x | = - x if x < 0

4) The equation | x | = k with k < 0 has no real solutions.

5) The equation | x | = k , k ≥ 0 is equivalent to x = k or x = - k

Solve the equation and check the answers found.

- If |x + 6 | = 7, then (see rule 5 above)

a) x + 6 = 7

or

b) x + 6 = -7

- Solve equation a)

x + 6 = 7

x = 1

- Solve equation b)

x + 6 = -7

x = -13

- solution x = 1

Left Side of Equation for x = 1.

|1 + 6 |

= | 7 |

= 7

Right Side of Equation for x = 1.

7 - x = -13

Left Side of Equation for x = 1.

|-13 + 6 |

= | -7 |

= 7

Right Side of Equation for x = 1.

7

The solutions to the given equation are x = 1 and x = -13

Solve the equation and check the answers found.

- Given

-2 |x / 2 + 3 | - 4 = -10

- We first write the equation in the form | A | = B. Add 4 to both sides and group like terms

-2|x / 2 + 3 | = -6

- Divide both sides by -2

|x / 2 + 3 | = 3

- We now proceed as in example 1 above, the equation

|x / 2 + 3 | = 3 gives two equations.

a) x / 2 + 3 = 3

or

b) x / 2 + 3 = -3

- Solve equation a)

x / 2 + 3 = 3

- to obtain

x = 0

- Solve equation b)

x / 2 + 3 = -3

- to obtain

x = -12

- x = 0

Left Side of Equation for x = 0.

-2 |x / 2 + 3 | - 4

= -2| 3 | - 4

= -10

Right Side of Equation for x = 1.

-10 - x = -12

Left Side of Equation for x = -12.

-2 |x / 2 + 3 | - 4

= -2 |-12 / 2 + 3 | - 4

= -2 |-6 + 3 | - 4

= -2(3) - 4

= -10

Right Side of Equation for x = -12.

-10

Solve the equation and check the answers found.

- If 2 x - 2 ≥ 0 which is equivalent to x ≥ 1, then |2 x - 2 | = 2 x - 2 (see rule 2 above) and the given equation becomes

2 x - 2 = x + 1

- Add 2 - x to both sides

x = 3

- Since x = 3 satisfies the condition x ≥ 1, it is a solution.
- If 2x - 2 < 0 which is equivalent to x < 1, then |2 x - 2 | = - (2 x - 2) (see rule 3 above) and the given equation becomes

-(2 x - 2) = x + 1

- Solve for x to obtain

x = 1 / 3

- Since x = 1 / 3 satisfies the condition x < 1, it is a solution.

- x = 3

Left Side of Equation for x = 3.

|2 x - 2 |

= |2*3 - 2 |

= 4

Right Side of Equation for x = 3.

x + 1

= 3 + 1

= 4 - x = 1/3

Left Side of Equation for x = 1 / 3.

|2 x - 2 |

= |2*(1/3) - 2 |

= 4 / 3

Right Side of Equation for x = 1 / 3.

x + 1

= 4 / 3

Solve the equation and check the answers found.

- If x
^{2}- 4 ≥ 0 ,or x^{2}≥ 4, then | x^{2}- 4 | = x^{2}- 4 and the given equation becomes

x^{2}- 4 = x + 2

- Add - (x + 2) to both sides

x^{2}- 4 -( x + 2) = 0

- Factor the left term

(x - 2)(x + 2) -( x + 2) = 0

(x + 2)(x - 2 -1) = 0

(x + 2)(x - 3) = 0

- Using the factor theorem, we can write two simpler equations

x + 2 = 0

or

x - 3 = 0

- Solve the above equations for x to find two values of x that make the left side of the equation equal to zero.

x = -2 and x = 3.

- Both values satisfy the condition x
^{2}≥ 4 and are solutions to the given equation.

x = -2 and x = 3.

- If x
^{2}- 4 < 0 ,or x^{2}< 4, then | x^{2}- 4 | = -(x^{2}- 4) and the given equation becomes.

-(x^{2}- 4) = x + 2

-(x^{2}- 4) - ( x + 2) = 0

- Factor the left term.

-(x - 2)(x + 2) - ( x + 2) = 0

(x - 2)(x + 2) + ( x + 2) = 0

(x - 2)(x + 2) + ( x + 2) = 0

(x + 2)(x - 2 + 1) = 0

(x + 2)(x - 1) = 0

- Two values make the left side of the above equation equal to zero

x = -2 and x = 1.

- Only x = 1 satisfies the condition x
^{2}< 4

- x = -2

Right Side of Equation = | x^{2}- 4 |

= | (-2)^{2}- 4 | = 0

Left Side of Equation = x + 2 = -2 + 2 = 0

- x = 3
Left Side of Equation = | x
^{2}- 4 |

= | 3^{2}- 4 |

= | 5 |

= 5 Right Side of Equation = x + 2 = 3 + 2 = 5 - x = 1

Left Side of Equation = | x^{2}- 4 |

= | 1^{2}- 4 | = | - 3 | = 3 Right Side of Equation = x + 2 = 1 + 2 = 3

The solutions to the given equation are x = -2, x = 1 and x = 3.

The above equation has two solutions

x = 2

x = -18

The above equation has two solutions

x = 3

x = -7

The above equation has two solutions

x = 0

x = -16/3

The above equation has one solution

x = 4

Solve the following absolute value equations

a) | x - 4 | = 9

b) | x

c) | x

d) | x + 1 | = x - 3

e) | -x | = 2

a) -5 , 13

b) -1 , 1

c) -3 , 2 , 4

d) no real solutions

e) -2 , 2

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