Graphing Inequalities

We have already graphed inequalities on a number line but what is the procedure and outcome of graphing inequalities on a coordinate plane?

As with number line graphs, we need to simplify the inequality expression before we try to graph.


Therefor: 2y > 4x - 6 must be simplified in the same manner as an equaiton.

We need y to have a coefficient of 1 so would divide each term in the inequality by 2.

2y > 4x - 6
2


y > 2x - 3

We would then graph the inequality the same way we graph equations, beginning with the y-intercept (-3) and then using the slope to extend the line in two directions.

Negative Exponents

Negative exponents have nothing to do with negative numbers.

If we think of the use of negative exponents in Scientific Notation we can understand them a bit better.



A negative exponent with a power of 10 means it is a decimal.




10 to the -3 = 0.001, which we can also write as 1 over 1000.





When we use powers of 10 in Scientific Notation we are changing the negative power to a decimal and multiplying it by the significant digits.




Significant digits = 2.63

10 to -3 = 0.001

Multiply and you have 0.00263









A negative exponent means "the reciprocal"


2 to the -3 can be rewritten as 1 over 2 to the 3rd (or 2 cubed).

2 cubed is 8.

Our final answer is one over eight.





When we are simplifying a more complex problem that includes a negative exponent we:





reduce any fractions

combine the exponets for each base:

x
and
y

Finally put any negative exponents in the opposite area of the fraction:


If there is a negative in the numerartor -> put it in the denominator.

If there is a negative in the denominator -> put it in the numerator.

Your final answer should never include a negative exponent.

Graphing the Solution to Multiple Inequalities

In an earlier chapter we learned how to graph an inequality on a number line. Now we look at graphing inequalties on the coordinate plane.

To graph an inequalitiy we will use the same procedure as graphing an equation.

Simplify the inequality first then use the slope and intercept to graph.

2y > 4x - 6


2y > 4x - 6
2

y > 2x - 3

We anchor our line with the y-intercept and use the slope to extend the line in both directions.

With inequalities we use two different forms of a line:

dash or broken line for greater than or less than
or
solid line for great than or equal to or less than or equal to

But with inequalities we also shade in the section of the coordinate plane that includes all the possible values for y

If y is greater than 2x - 3 we shade in above the line.

For y is less than -x + 4 we shade below the line.

When we graph two inequalities we shade each individually and the area where they overlap is the solution to both inequalities

Important steps: simplify the inequality, graph the line using y-intercept and slope with either a dashed or solid line, shade above for greater and below for less.