Question

How do you solve 2x+3>7 or 2x+9>11?

Answer 1

The solution set is `{x | x>1}`.

Explanation:

For each of these inequalities, there will be a set of `x`-values that make them true. For example, it's pretty clear that large values of `x` (like 1,000) work for both, and negative values (like -1,000) will not work for either.

Since we're asked to solve a "this OR that" pair of inequalities, what we'd like to know are all the `x`-values that will work for at least one of them. To do this, we solve both inequalities for `x`, and then overlap the two solution sets.

Inequality 1:

`2x+3>7" "=>" "2x>4"    "`(subtract 3 from both sides)
`color(white)(2x+3>7)" "=>"     "x>2"    "`(divide both sides by 2)

Inequality 2:

`2x+9>11" "=>" "2x>2"    "`(subtract 9 from both sides)
`color(white)(2x+9>11)" "=>"     "x>1"    "`(divide both sides by 2)

So we need to list all the `x`-values that satisfy either `x>2` or `x>1`.

In this case, if an `x`-value is greater than 2, it will automatically be greater than 1. Thus, the solution set for `2x+3>7` is a subset of the one for `2x+9>11`. That means, all we need to do here is list the solution set for `2x+9>11`, and we're done.

The solution set we need is simply "all `x` such that `x` is greater than 1", or (in math terms):

`{x | x>1}`
or
`x in (1, oo)`