Difference Between Linear Equations & Linear Inequality
In mathematics, a Cartesian plane is used to map out algebraic functions, which include lines, line segments, plots of points, arcs and polygons. This plane uses a grid with an "x" axis and a "y" axis. Starting from the intersection of these axes, the locations of points are defined with positive and negative numbers that represent their distance from the central point, or "origin." Linear equations and linear inequalities both deal with lines and spaces that are plotted on the Cartesian plane in this manner.
Other People Are Reading
-
Linear Equation Basics
-
A linear equation is generally represented by y=mx+b. This equation shows the location and slope of a line by giving a constant for the measurement of the "y" (vertical) coordinate in relation to a given "x" (horizontal) coordinate. In this way, "m" defines the slope of the line, while "b" defines an additional distance from the origin. In order to plot a linear equation, you need values for "m" and "b," because these are the variables that dictate what the "y" value will be for a given "x" value on the line or vice versa.
Example of a Linear Equation
-
For the equation y=2x, "b" will be equal to zero. This means that, regardless of slope (the value of "m"), the line will cross through the origin. Since the value for "m" is 2, the value for a given "x" coordinate will be double whatever the value for the "y" coordinate is. For the equation y=2x+3, the line will look exactly the same, only it will be 3 units higher on the "y" axis, with the line passing through the "y" axis at 3 instead of zero.
-
Linear Inequality Basics
-
A linear inequality is the same as a linear equation, except that it represents an area greater or lesser than the values of a specific line rather than the line itself. This is done by using the ">," "<," "≤" or "≥" symbol instead of the "=" symbol.
Example of a Linear Inequality
-
For y>2x+3, you would chart a line exactly as you would for y=2x+3, except that the line would be represented by a dotted line instead of a solid line, and the area above the line will be shaded. Likewise, for y<2x+3, the dotted line will fall in exactly the same place, only the area below the dotted line will be shaded instead of the area above it. When a "≥" or "≤" is used, you would do the same thing as you would for ">" or "<," respectively, except you would draw a solid line instead of a dotted line. This shows that the "y" value can be equal to the value of the actual line, while the ">" and "<" symbols show that the "y" is less than or more than the value of the line, but never equal to it.
Vertical Lines
-
With both linear equations and linear equalities, vertical lines cannot be represented with the equation y=mx+b. Instead, you must represent it with the equation x=b, with "b" representing the point where the line crosses the "x" axis. This is because the line has a slope that cannot be defined. This is true for both linear equations and linear equalities, except that linear equalities will again use the ">," "<," "≤" or "≥" symbols instead of the "=" symbol, having dotted lines for the inequalities with ">" and "<," and solid lines for the equations with "≤" and "≥," and you will shade an area to the right or left of the solid or dotted line.
-
References
- Photo Credit Mathematik image by bbroianigo from Fotolia.com
