Question

Solve (x - 2 < 5) ∩ (x + 7 > 6).

Answer 1

-1<x<7

Explanation:

We need to solve each inequality and then from the interval of numbers they both have in common (doing this because of the 'and').

x-2<5

Add 2 on both sides:

x<5+2

Simplify:

x<7

These are values of x that are less than 7.

x+7>6

Subtract 7 on both sides:

x>6-7

Simplify:

x>-1

These are values of x greater than -1.

So the values they have in common are the numbers in between -1 and 7.

That is x<7 and x>-1.

You can also write it as -1<x<7.

Maybe you need a graph to convince you more.

○~~~~~~~~~~~~~~~~~~~~~~~~~~~

~~~~~~~~~~~~~~~~~~~~~~~~~○

‐-------------(-1)------------------------(7)-----------------

So where you see both graphs is the solution:

○~~~~~~~~~~~~~~○

‐-------------(-1)------------------------(7)-----------------

Answer 2

For this case we must solve the following expression:

`x-2 <5` intersected
`x + 7> 6`

We have:

`x-2 <5`

Adding 2 to both sides of the inequality:

`x <5 + 2\\x <7`

The solution is given by all values of x less strict than 7.

Now we have:

`x + 7> 6`

Subtracting 7 from both sides of the inequality:

`x> 6-7\\x> -1`

The solution is given by all values of x higher than -1.

If we intersect the solutions we have:

The solution is given by all values of x between -1 and 7. That is:

(-1,7)

Answer:

`-1 <x <7`