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## Give Solutions to Inequality

#### Introduction

When you solve an inequality, you get a range of solutions. If you get the equation $x < 8$, the answer can be any number less than 8. So, dealing only with integers, the solutions are $7,6,5,4,3,2,1,0,-1,-2,-3$ and so on.

To give the solutions to an inequality we need to use Interval Notation, which is a way of showing a range using only the ends of the interval.

## Lesson

To give a range of solutions to an inequality, we first have to solve the equation.

If we're given:

$-4 - 2 < x ≤ 5 + 11$

We would first combine the like terms to get:

$-6 < x ≤ 16$

Therefore the solutions to the problem are any number greater than -6 (but not -6 itself) and any number smaller than 16 (which could include 16). In other words, the integer solutions are:

$-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16$

To write this out with interval notations we would set the intervals as -6 and 16. Because the solution has to be greater than, but can't equal, -6 we use the curved bracket; because it can equal 16 we use a square bracket. That gives us:

$(-6, 16]$

If we take another equation:

$x + 2 < 5$

We would subtract 2 from both sides to isolate the x:

$x + 2 (-2) < 5 (-2)$

$x < 3$

So, the solutions to the equation are any numbers greater than 3 (but not 3 itself).

Since just above 3 is our lowest interval, we use a curved bracket and the number (3. The highest possible solution is infinite, so we use the infinity symbol and a square bracket ∞].

Therefore, the range of solutions is (3,∞].