|X+4| = |3X+10|

Answer 1

first conditions:
x+4 ≥0 ---> x ≥ -4
and
3x+10 ≥0 ----> x ≥-10/3
so we know that x ≥ -10/3

case 1:
x+4 = 3x+10
-2x = 6
x = -3

case 2:

-x-4 = 3x+10
-4x = 16
x = -4 , also no good, because of above restriction

So x = -3

check:
if x = -4
LS = |-4+4| = 0
RS = |-12+10| = 2
LS ≠ RS

if x = -3
LS = |1| = 1
RS = |-9+10| = 1

So, x = -3

Answer 2

| x + 4 | = | 3 x + 10 |

x + 4 = 3 x + 10

OR

x + 4 = - ( 3 x + 10 )

1 )

x + 4 = 3 x + 10 Subtract x to both sides

x + 4 - x = 3 x + 10 - x

4 = 2 x + 10 Subtact 4 to both sides

4 - 4 = 2 x + 10 - 4

0 = 2 x + 6 Subtract 2 x to both sides

0 - 2 x = 2 x + 6 - 2 x

- 2 x = 6 Divide both sides by - 2

- 2 x / - 2 = 6 / - 2

x = - 3

2)

x + 4 = - ( 3 x + 10 )

x + 4 = - 3 x - 10

Subtract 4 to both sides

x + 4 - 4 = - 3 x - 10 - 4

x = - 3 x - 14 Add 3 x to both sides

x + 3 x = - 3 x - 14 + 3 x

4 x = - 14 Divide both sides by 4

4 x / 4 = - 14 / 4

x = - 14 / 4 = - 2 * 7 / ( 2 * 2 ) = - 7 / 2

The solutions are :

x = - 7 / 2 and x = - 3

Proof :

For x = - 7 / 2

| x + 4 | = | 3 x + 10 |

| - 7 / 2 + 4 | = | 3 * ( - 7 / 2 ) + 10 |

| - 7 / 2 + 8 / 2 | = | - 21 / 2 + 10 |

| 1 / 2 | = | - 21 / 2 + 20 / 2 |

| 1 / 2 | = | - 1 / 2 |

1 / 2 = 1 / 2

For x = - 3

| x + 4 | = | 3 x + 10 |

| - 3 + 4 | = | 3 * ( - 3 ) + 10 |

| 1 | = | - 9 + 10 |

| 1 | = | 1 |

1 = 1