Question

Help me. Please give explanation. Thanks

Answer 1

Answer: x < 3

Explanation:

NOTE:
`√((x+5)^2) = |x + 5|`

Rewrite the equation:

| x + 5 | < | x + 1 |

Remember that absolute value makes the "inside" value positive, so we need to solve for 2 case of x + 5 (inside is positive or negative) and 2 cases of x + 1.

± (x + 5) < ± (x + 1)

Case 1: + +

x + 5 < x + 1

5 < 1

FALSE

Case 2: - -

-(x + 5) < -(x + 1)

x + 5 > x + 1 divided both sides by -1

5 > 1

TRUE so this works when both "insides" are negative

⇒ x + 5 < 0 and x + 1 < 0

x < -5 and x < -1

since it has to satisfy both, x < -5

Case 3: + -

x + 5 < -(x + 1)

x + 5 < -x - 1 distributed -1 on right side

2x + 5 < -1

2x < -6

x < -3

Case 4: - +

-(x + 5) < x + 1

-x - 5 < x + 1 distributed -1 on left side

-5 < 2x + 1

-6 < 2x

-3 < x

x > -3

This is a contradiction of Case 3 so need to choose a test point.

TEST POINTS (I choose -6, -4, and -2)

| -6 + 5 | < | -6 + 1 |

| -1 | < | -5 |

1 < 5

TRUE so x < -5 is valid

| -4 + 5 | < | -4 + 1 |

| 1 | < | -3 |

1 < 3

TRUE so x < -3 is valid

| -2 + 5 | < | -2 + 1 |

| 3 | < | -1 |

3 < 1

FALSE so x > 3 is NOT valid

Note that x < -3 includes all the values of x < -5, so x < -3 is the answer.

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This is a faster way to solve it. Thank you @proner26 !

| x + 5 | < | x + 1 |

(| x + 5 |)² < (| x + 1 |)²

x² + 10x + 25 < x² + 2x + 1

10x + 25 < 2x + 1

8x + 25 < 1

8x < -24

x < -3

Answer: x < -3