graphing inequailities

Question

compare and contrast two variable equations and two variable inequailities address the following aspects of each what is the same/different about how they look in algebraic form? what is the same/different about their graphs? how is the process of graphing them the same as or different from each other? how does the number of solutions to a system of equations compare to the number of solutions to a system of inequailities? a worked example of how to graph a two variable inequailty on the coordinate plane. you can attach the graph if necessary.

Answer 1

A 2-variable equation (let the variables be x and y, where x is the independent variable) in its simplest form would be represented graphically by a sloping straight line. All the points on the line represent ordered pair solutions of the linear (straight line), that is, all the paired coordinates (x,y) which satisfy the linear equation, given by y=mx+a where m is the slope of the line (the tangent of the inclination of the line to the horizontal) and a is the y-intercept, where the line intercepts the y-axis (vertical).

A linear inequality starts with the same line as a linear equation, but this line represents a boundary between what satisfies the inequality and what doesn't. The first type of inequality is not-equals (≠). This is represented by all points that don't lie on the line, so it's the two areas on either side of the line.

Then we have less than (<) or greater than (>). Not-equals can often be written <>, that is, < or >. So it's represented by the area beneath the line (<) or above the line (>), but not on it.

Finally, we have less-than-or-equal-to (≤) or greater-than-or-equal-to (≥), which is like < or > but also includes points on the line. ≤ and ≥ overlap because they each include the line itself.

Graphically, inequalities are shown by shading the relevant area (actually in practical terms, part of the area, because the line has infinite length, making the areas infinite in size). When the line itself is excluded it's normally shown as a dotted or broken line.

If either of the two variables has a higher degree than 1 (the exponent indicates the degree) then we no longer have a straight line, but a curve, and some curves can be closed, like circles and ellipses. The same rules apply when comparing equations with inequalities. For example, x²+y²<r² is represented by the inside area of a circle (excluding the circle itself) and x²+y²>r² by the whole area outside the circle.

The curve represents a boundary in the case of inequalities.

Because inequalities are represented by areas, all points within the area satisfy the inequality so there are an infinite number of points, that is, an infinite number of solutions to the inequalities, shown by indicating ranges of the variable pairs.