Question

How do you write the inequality and solve given "twice a number increased by 3 is less than the number decreased by 4"?

Answer 1

See a solution process below:

Explanation:

To write the inequality we will start by defining "a number" as `n`.

Then, "twice a number" can be written as `2n`.

"increased by 3" becomes: `2n + 3`

"is less than" turns it into" `2n + 3 <`

"the number decreased by 4" is: `n - 4`

Putting this together gives the inequality:

`2n + 3 < n - 4`

To solve, first subtract `color(red)(3)` and `color(blue)(n)` from each side of the inequality to solve for `n` while keeping the inequality balanced:

`-color(blue)(n) + 2n + 3 - color(red)(3) < -color(blue)(n) + n - 4 - color(red)(3)`

`-color(blue)(1n) + 2n + 0 < 0 - 7`

`(-color(blue)(1) + 2)n < -7`

`1n < -7`

`n < -7`

Answer 2

Any number less than `-7` will make this inequality true.

`x < -7`

Explanation:

Let the number be `x`

Write the left side as maths first:

`color(red)("Twice a number ") color(blue)(" increased by 3 ") color(magenta)("is less than")`

`color(red)(2x) color(white)(wwwwwwwwwwwww)color(blue)(+ 3 ) color(white)(wwwwwww)color(magenta)( <)`

Now do the same for the right side..

`color(red)(2x)" " color(blue)(+ 3 ) " "color(magenta)( <)" "color(green)("the number decreased by 4"`

`color(red)(2x)" " color(blue)(+ 3 ) " "color(magenta)( <)" "color(green)(x-4)`

We have the inequality, now solve it:

`2x +3 < x-4`

`2x-x < -4 -3`

`x < -7`