Solve the inequality 2|43x − 5| + 2 ≤ 7. Write your answer in interval notation.

Answer 1

2| 43x - 5 | <= 5

| 43x - 5 | < = 5/2

43x - 5 <= 5/2 or 43x - 5 >= -5/2

43x <= 5/2 + 5 43x > = -5/2 + 5

43x <= 5/2 + 10/2 43x >= -5/2 + 10/2

43x <= 15/2 43x >= 5/2

x <= 15/86 x>=5/86, which is

the same as 5/86 <=x

putting them together:

5/86 <= x <= 15/86

check:

low endpoint

x=5/86 --> 2 | 43 * 5/86 - 5 |

= 2 | 5/2 - 5 |

= 2 | 5/2 - 10/2 |

= 2 | -5/2 |

= 2 (5/2) = 5, so yes it checks

upper endpoint

x = 15/86

2 | 43(5/86) - 5 |

2 | (5/2) - 5 |

2 | 5/2 - 10/2 |

2 | -5/2 |

2 (5/2) = 5, so yes it checks

in between:

x = 10/86

2| 43(10/86) - 5 |=

2 | 10/2 - 5 | =

2| 5 -5 | =0 <=5, so yes it checks

too small:

x=0

2 | 0 - 5| = 2(5) = 10 <=5

Fails as expected

too big:

x=1

2|43-5| = 76 <= 5

Fails as expected

PROVEN answer:

5/86 <= x <= 15/86

Answer 2

2|43x - 5| + 2 ≤ 7

2|43x - 5| ≤ 5 (subtract 2 from each side)

|43x - 5| ≤ 2.5 (divide each side by 2

SO

-2.5 ≤ 43x - 5 ≤ 2.5

2.5 ≤ 43x ≤ 7.5 (add 5 to each part of the expression)

2.5/43 ≤ x ≤ 7.5/43 (divide each part of the expression by 43)

25/430 ≤ x ≤ 75/430

5/86 ≤ x ≤ 15/86

In interval notation, the answer would be [5/86, 15/86]