Double Inequalities

Double inequalities on a system, that is which has the two inequalities with associated to one variable. It consists of each and every one values of the variable that makes each one of the inequality in the system be true. A system in which it has f (x) ≥ a, and f (x) ≤ b, where as the same expression appears on both inequalities. It is commonly called a double inequality and is often written in the form a ≤ f (x) ≤ b. Be sure that both inequality signs are pointing in the same direction and that the double inequality is only used to show an expression in x attentive between two values. It must be also a should be less than to the value or equal to b in the inequalities function.

a ≤ f (x) ≤ b or b ≥ f (x) ≥ a.
Here we are going to solve the double inequalities and the problems related to the double inequalities.

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Basic Rules of inequalities:

a < b, it means a is smaller than b.
a `<=` b, it means a is smaller than or equal to b.
a > b, it means a is larger than b,
a `>=` b, it means a is larger than or equal to b.
a `!=` b, it means a is smaller than b or larger than b but not equal to b.



Mathematical operations used in the inequalities Math 1:

Addition:

If a < b , then in addition c on both sides a + c < b + c. same for a > b, a `<=` b, a`>=` b, a `!=` b.

Subtraction:

If a < b , then in subtracting c on both sides a - c < b - c. same for a > b, a `<=` b, a`>=` b, a `!=` b.

Multiplication by positive c:

If a < b, then multiplying with positive c on both sides ac < bc. as same for a > b, a `<=` b, a`>=` b, a `!=` b.

Multiplication by negative c:

If a < b, then multiplying with negative c on both sides ac > bc. as same for a > b, a `<=` b, a`>=` b, a `!=` b.

Divide by Positive c:

If a < b, then dividing with positive c on both sides `a/c` < `b/c` . as same for a > b, a `<=` b, a`>=` b, a `!=` b.

Divide by negative c:
If a < b, then dividing with negative c on both sides `a/c` > `b/c` . as same for a > b, a `<=` b, a`>=` b, a` !=` b.


Double Inequalities - Example Problems:

Double Inequalities - Problem 1:

Solve ` 12 >= x + 18 >= 30`

Solution:

` 12 >=x + 18 >= 30 `

Add with  -12 on both sides

`12-12>=x + 18 -12 >= 30 -12 `

`0 >= x + 6 >= 18`

Add with - 6 on either sides

`0 -6 >= x >= 18 -6`

`-6 >= x >= 12`

Double Inequalities -  Problem 2:

Solve ` 24 <= -x+16 <= 36`

Solution:

`24 <= -x + 16 <= 36`

Add with - 24 on both sides

`24 - 24 <= -x+16- 24 <= 36 -24 `

`0 <= -x - 8 <= 12 `

Add with +8 an either sides.

`0 +8 <= -x -8+8 <= 12 + 8 `

`8 <= -x <= 20`

Divide by -1 an either sides

`((8)/(-1)) <= ((-x)/(-1)) <= (20/(-1))`

Due to the change in negative sign the inequalities symbol also changes.

`-8 gt= x gt= -20`

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Double Inequalities - Practice Problem:

Problem 1:

Solve ` 16 <= -x+24 <= 48`

Answer:`8 >= x >= -24`

Problem 2:

Solve ` 15 >= x+20 >= 25`

Ansewer: `10 >= x >= 20 `