Algebra/Inequalities

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Graphs of inequalities in one variable. The first number line shows x>3, x<-7, and -4<x≤0. The second line shows the disjunction x<-2 or x>2.

Inequalities, unlike equations, commonly have an infinite number of solutions. Inequalities can be expressed in 1 or more variables.

means that x is greater than A

means that x is less than A

means that x is greater than or equal to A

means that x is less than or equal to A

[edit] Inequalities in 1 variable

[edit] Solving them

For example, you have the inequality . You can solve for x just like a normal equation. Adding 3 to both sides yields .

[edit] Special Cases - A variable on the denominator

For example consider the ineqaulity

In this case one cannot multiply the right hand side by (x-1) because the value of x is unknown. Since x may be either positive or negative, you can't know whether to leave the inequality sign as <, or reverse it to >. The method for solving this kind of ineqaulity involves four steps:

1. Find out when the denominator is equal to 0. In this case it's when x = 1.
2. Pretend the inequality sign is an = sign and solve it as such: , so x = 2.
3. Plot the points x = 1 and x = 2 on a number line with an unfilled circle because the original equation included < (it would have been a filled circle if the original equation included <= or >=). You now have three regions: x < 1, 1 < x < 2, and x > 2.
4. Test each region independently. in this case test if the inequality is true for 1<x<2 by picking a point in this region (e.g. x=1.5) and trying it in the orgiinal inequation. For x=1.5 the original inequation dosn't hold. So then try for 1>x>2 (e.g. x=3). In this case the original inequation holds, and so the solution for the original inequation is 1>x>2.

[edit] Inequalities in 2 variables

A graph of an inequality in 2 variables.

Linear inequalities in 2 variables are typically in the form of , where m is the slope of the line and b is the y-intercept.

Graphing an inequality is easy. First, graph the inequality as if it were an equation. If the sign is ≤ or ≥, graph a normal line. If it is > or <, then use a dotted or dashed line. Then, shade either above or below the line, depending on if y is greater or less than mx + b.