Systems of Inequalities

1. Systems of inequalities let you describe many possible solutions at once.
2. Instead of finding a single point (as with a system of equations), you find a region of points that satisfy all given inequalities simultaneously.
3. This is a powerful way to build and test models for real decisions.

Inequalities and Solution Sets

A linear inequality looks like a linear equation but uses an inequality sign, for example: $$2 x-3>5, \quad 4 y+7 \leq 19, \quad 3 x-2 y \geq 6$$

Its solution set is the set of all values that make the statement true.

1. For one variable, the solution set is usually an interval on a number line (for example, $x < 4$).
2. For two variables (like $x$ and $y$), the solution set is typically a shaded region in the coordinate plane.

Equivalence Transformations

1. To solve inequalities algebraically, you use equivalence transformations, similar to solving equations.
2. The key idea is that you want to transform the inequality into an equivalent one with the same solution set.
3. For real numbers $a$ and $b$ :
   1. $a>b$ if and only if $a-b>0$.
   2. $a<b$ if and only if $a-b<0$
4. We can transform inequalities, just like equations, as long as we keep the same solution set.

Theorem 1 – Adding/Subtracting the Same Number

Adding or subtracting the same real number on both sides of an inequality gives another equivalent inequality.

1. If $a>b$, then for any real $c$:
   1. $a+c>b+c$
   2. $a-c>b-c$
2. and similarly, if $a<b$, then
   1. $a+c<b+c$
   2. $a-c<b-c$

Theorem 2 – Multiplying by a Positive Number

Multiplying both sides of an inequality by the same positive real number leaves the direction the same.

1. If $a>b$ and $k>0$, then $a k>b k$.
2. If $a<b$ and $k>0$, then $a k<b k$.

Theorem 3 – Multiplying by a Negative Number

Multiplying or dividing both sides of an inequality by a negative real number reverses the direction of the inequality.

1. If $a>b$ and $k<0$, then $a k<b k$.