Question

How do you solve and write the following in interval notation: `-5(13x + 3) < - 2(13x - 3)`?

Answer 1

See a solution process below:

Explanation:

First, expand the terms on each side of the inequality by multiplying each term within the parenthesis by the term outside the parenthesis:

`color(red)(-5)(13x + 3) < color(blue)(-2)(13x - 3)`

`(color(red)(-5) xx 13x) + (color(red)(-5) xx 3) < (color(blue)(-2) xx 13x) - (color(blue)(-2) xx 3)`

`-65x + (-15) < -26x - (-6)`

`-65x - 15 < -26x + 6`

Next, add `color(red)(65x)` and subtract `color(blue)(6)` from each side of the inequality to isolate the `x` term while keeping the inequality balanced:

#color(red)(65x) - 65x - 15 - color(blue)(6) < color(red)(65x) - 26x + 6 - color(blue)(6)#

`0 - 21 < (color(red)(65) - 26)x + 0`

`-21 < 39x`

Now, divide each side of the inequality by `color(red)(39)` to solve for `x` while keeping the inequality balanced:

`-21/color(red)(39) < (39x)/color(red)(39)`

`-(3 xx 7)/color(red)(3 xx 13) < (color(red)(cancel(color(black)(39)))x)/cancel(color(red)(39))`

`-(color(red)(cancel(color(black)(3))) xx 7)/color(red)(color(black)(cancel(color(red)(3))) xx 13) < x`

`-7/13 < x`

To state the solution in terms of `x` we can reverse or "flip" the entire inequality:

`x > -7/13`