System of Absolute Values!

Algebra Level 1

If the solution set of the system of inequalities $\begin{cases}\lvert a+1 \rvert < 3\\ \lvert b-1 \rvert <10 \end{cases}$ is $x < a+b < y,$ then what are $x$ and $y?$

x={-12}, y={14}

x={-12}, y={12}

x={-13}, y={13}

x={-13}, y={14}

1 solution

Eli Ross Staff
Nov 4, 2015

From the first inequality, we have $\begin{aligned} -3 &< a+1 < 3 \\ -4 &< a < 2. \qquad (1) \end{aligned}$

From the second inequality, we have $\begin{aligned} -10 &< b-1 < 10 \\ -9 &< b < 11. \qquad (2) \end{aligned}$

Then from $(1)$ and $(2),$ we have $\begin{array} { r r r } & -4 < a < 2 \\ + & \ -9 < b < 11 \\ \hline & -13< a+b < 13. \end{array}$

Thus, the answers are $x=-13$ and $y=13.$

The solution given is incorrect. From the first inequality we get a = - 3 to +1 From the second inequality we get b = - 8 to +10 a + b can have a maximum value of + 11 and minimum - 11. Hence x = - 12 and y = +12 is the most appropriate answer. To validate your answer (of +13 and - 13) please try to show a condition where a + b can be + 12 or - 12.

Ramachandran Nair - 5 years, 5 months ago

I agree with Mr. Nair. -12 to +12.

CHARLES GALLAGHER - 5 years, 5 months ago

Take $a = 1.9$ and $b = 10.9$ . These satisfy the inequalities, yet $a+b = 12.8 > 12.$ Keep in mind the numbers do not have to be whole numbers.

Eli Ross Staff - 5 years, 5 months ago

$a = 1.5, b= 10.5$

Eli Ross Staff - 5 years, 5 months ago

i agree that x = -13 but y must be greater then 13 because the inequality here is < then not <= to y so y is 14

Ridha Omri - 5 years ago

$y$ cannot be $\ge 13.$ Note that $a<2$ and $b<11,$ so $a+b < 13.$

Eli Ross Staff - 5 years ago

how can we solve this question???? If LaTeX: $|x+1| \leq 9$ and LaTeX: $|y-1| \leq 4$ , then LaTeX: $a \leq 3y-2x \leq b$ . What is the value of LaTeX: $b-a$ ?

PARTH GUPTA - 4 years, 2 months ago

i agree, i had a hard time finding the correct answer.

Helen Wang - 1 year, 9 months ago

System of Absolute Values!

1 solution

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