Inequalities When To Flip Sign

Multiplying & Dividing Inequalities

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Definitions
Operating With Inequalities: Multiplying & Dividing
1. The Exception: Negative Numbers
Multiple Inequalities

Definitions

An inequality compares two values.

Inequality - A comparison of 2 values or expressions.
For example, 10x < 50 is an inequality whereas x = 5 is an equation.
Equation - A statement declaring the equality of 2 expressions.
For example, 4x = viii is an equation whereas 10x > 20 is an inequality.

Operating With Inequalities: Multiplying & Dividing

Performing multiplication or sectionalization with an inequality is nigh identical to multiplying or dividing parts of traditional equations (with one exception, covered beneath).

Consider the following examples:

10x + 15 < 25 + 5x
10x + xv - 15 < 25 - 15 + 5x
10x < x + 5x
10x - 5x < 10 + 5x - 5x
5x < 10
10 < 2

The Exception: Negative Numbers

There is 1 very of import exception to the dominion that multiplying or dividing an inequality is the aforementioned every bit multiplying or dividing an equation.

Whenever you multiply or split an inequality by a negative number, y'all must flip the inequality sign.

In the post-obit example, detect how the < sign becomes a > sign when the inequality is divided past -two

-2x - 10 < 2
-2x - 10 + 10 < 2 + 10
-2x < 12
x > -6 [Dividing by -2 required the flipping of the inequality sign]

In the following example, notice how the < sign becomes a > sign when the inequality is divided by -2

-2x + fifteen < three
-2x + 15 - 15 < 3 - 15
-2x < -12
ten > 6 [Dividing by -two required the flipping of the inequality sign]

Alert: Caution When Multiplying or Dividing Variables

One very important implication of this rule is: You cannot divide by an unknown (i.e., a variable) unless you are sure of its sign since you do not know whether you must flip the sign of the inequality. There are plenty of instances where you will know the sign of a variable and equally a consequence, you can multiply or divide and know for sure whether you must flip the inequality sign. However, yous must always ask yourself whether you know for sure the sign of the variable before dividing or multiplying when dealing with an inequality.

If 2x5y < 10y, what is the range of potential values for x?
Y'all cannot split up by y or 5y since yous practise non know whether y is negative or positive and, as such, you do not know whether to flip the inequality.

Multiple Inequalities

Just every bit it is possible to solve two simultaneous equations, so information technology is possible to solve ii inequalities (or three, or four, etc.). In solving multiple simultaneous inequalities using multiplication or sectionalisation, the about important office is to solve each inequality separately and so combine them.

If 2x < 10, -5x < -10, and 15x < 150, what is the range of possible values for x?

1.) Solve each inequality lone.
2x < 10
ten < 5 [Notation: The inequality is not flipped since we are dividing by 2, which is positive]

-5x < -10
10 > 2 [Since we dissever by -v, a negative number, we flip the inequality sign]

15x < 150
x < 10

2.) Combine each inequality and find the overlap (i.e., the areas where each inequality is satisfied--this area is the solution).
x < 5
x > 2
ten < x

The area of overlap--i.e., the solution to the set of inequalities--is where x < 5 and x > ii

For many students, the higher up set of inequalities can best be understood graphically. The solution to the set of inequalities is the overlapping graphical area.