How do you evaluate 7| 10+ v | - 1\leq 90?

2 Answers
Sep 15, 2017

-23le v le3

Explanation:

7|10+v|-1le90
First, we will add 1 to both sides.
7|10+v|-1color(blue)+color(blue)1le90color(blue)+color(blue)1
This is also equal to
7|10+v|le91
Now we divide both sides by 7
(cancel7|10+v|)/cancel7lecancel91/cancel7 13
This means that |10+v|le13.
And it also means that 10-vge-13

10+vle13
Subtract 10 from both sides.
10+vcolor(blue)-color(blue)10le13color(blue)-color(blue)10
Simplify.
vle3

10+vge-13
Subtract 10 from both sides.
10+vcolor(blue)-color(blue)10ge-13color(blue)-color(blue)10
Simplify.
vge-23

Combine it together.
-23levle3

Sep 15, 2017

See a solution process below:

Explanation:

First, add color(red)(1) to each side of the inequality to isolate the absolute value term while keeping the inequality balanced:

7abs(10 + v) - 1 + color(red)(1) <= 90 + color(red)(1)

7abs(10 + v) - 0 <= 91

7abs(10 + v) <= 91

Next, divide each side of the inequality by color(red)(7) to isolate the absolute value function while keeping the inequality balanced:

(7abs(10 + v))/color(red)(7) <= 91/color(red)(7)

(color(red)(cancel(color(black)(7)))abs(10 + v))/cancel(color(red)(7)) <= 13

abs(10 + v) <= 13

The absolute value function takes any term and transforms it to its non-negative form. Therefore, we must solve the term within the absolute value function for both its negative and positive equivalent.

-13 <= 10 + v <= 13

Now, subtract color(red)(10) from each segment of the system of inequalities to solve for v while keeping the system balanced:

-color(red)(10) - 13 <= -color(red)(10) + 10 + v <= -color(red)(10) + 13

-23 <= 0 + v <= 3

-23 <= v <= 3

Or

v >= -23 and v <= 3

Or, in interval notation:

[-23, 3]