Inequalities

Inequality in mathematics, statement that a mathematical expression is less than or greater than some other expression; an inequality is not as specific as an equation equation, in mathematics, a statement, usually written in symbols, that states the equality of two quantities or algebraic expressions, e.g., x+3=5. The quantity x , but it does contain information about the expressions involved. The symbols < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to) are used in place of the equals sign in expressions of inequalities. As in the case of equations, inequalities can be transformed in various ways. The direction of the inequality remains unchanged if some number is added to both sides or subtracted from both sides or if both sides are multiplied or divided by some positive number; e.g., subtracting 10 from both sides of the inequality x <> −4.

Two Step Equations and Inequalities

It takes two steps to solve an equation or inequality that has more than one operation:

1.Simplify using the inverse of addition or subtraction.
2.Simplify further by using the inverse of multiplication or division.

Remember, when you multiply or divide an inequality by a negative number, you must reverse the inequality symbol.

Here's a two-step equation. Let's start with the variable x, and describe, step by step, what is being done to x in an equation.

3x - 10 = 14 Equation
3x First, x is multiplied by three.
3x - 10 Next, ten is subtracted from the term 3x.
3x - 10 = 14 We get a result of 14.

Start with x --> Multiply by 3 --> Subtract 10 -->Result is 14.

Solving an equation is like running the equation backwards to discover what number will work in the equation. Now let's work backwards and use inverse operations to undo all the steps. We can start with the result of 14.

14 Start with result.
14 + 10 Next, working backwards, we can add 10, which is the inverse of subtracting 10.

14 + 10/3 Now we divide by 3, since that's the inverse of multiplying by 3.
24/3 = 8 We get an answer of 8.

Start with result of 14 —> Add 10 —> Divide by 3. Answer is 8.

Do you see how it's important when solving an equation to "undo" all the steps in the correct order? No matter how many steps are in the original equation, you can work backwards and apply the inverse operations, in order, to arrive at the solution!

You can solve two-step inequalities in exactly the same way. Just work backwards, using the inverse operations, to arrive at the solution. But watch out when multiplying or dividing by a negative number! Just as you learned in the last lesson, you must reverse the inequality symbol.

Properties of Inequalities:


Addition property:

If x <> y, then x + z > y + z

Example: Suppose Sylvia's weight > Jennifer's weight, then Sylvia's weight + 9 > Jennifer's weight + 9

Or suppose 4 > 2, then 4 + 5 > 2 + 5


Subtraction property:

If x <> y, then x − z > y − z

Example: Suppose Sylvia's weight > Jennifer's weight, then Sylvia's weight − 9 > Jennifer's weight − 9

Or suppose 8 > 3, then 8 − 2 > 3 − 2


Multiplication property:

If x <> 0 then x × z < z =" 10"> 0)

If x > y, and z > 0 then x × z > y × z

Example: Suppose 20 > 10, then 20 × 2 > 10 × 2

If x <> y × z

Example: Suppose 2 <> 5 × -4 ( -8 > -20. z = -4 and -4 <> y, and z <> 1, then 5 × -2

Division property:

It works exactly the same way as multiplication

If x <> 0 then x ÷ z <> y, and z > 0 then x ÷ z > y ÷ z

Example: Suppose 20 > 10, then 20 ÷ 5 > 10 ÷ 5

If x <> y ÷ z

Example: Suppose 4 <> 8 ÷ -2 ( -2 > -4 )

If x > y, and z <> 1, then 5 ÷ -1

Transitive property:

If x > y and y > z, then x > z

Example: Suppose 10 > 5 and 5 > 2, then 10 > 2

Comparison property:

If x = y + z and z > 0 then x > y

Example: 6 = 4 + 2, then 6 > 4

Reference:
http://encyclopedia2.thefreedictionary.com/Inequality+%28mathematics%29
http://www.math.com/school/subject2/lessons/S2U3L6EX.html
http://www.basic-mathematics.com/properties-of-inequality.html