Here are the steps required for Solving Polynomial Inequalities:
Instead, we will mostly use the geometric definition of the absolute value: The absolute value of a number measures its distance to the origin on the real number line.
We are ready for our first inequality.
Obviously we are talking about the interval -5,5: What about the solutions to? On the left side, real numbers less than or equal to -2 qualify, on the right all real numbers greater than or equal to 2: We can write this interval notation as What is the geometric meaning of x-y?
Consider the example -4 Let's find the solutions to the inequality: Which real numbers are not more than 1 unit apart from 2?
We're talking about the numbers in the interval [1,3]. What about the example Let's rewrite this as which we can translate into the quest for those numbers x whose distance to -1 is at least 3.
We first divide both sides by 2. Note that absolute values interact nicely with multiplication and division: Thus we obtain after simplification, we get the inequality asking the question, which numbers are less than 1 unit apart from So the original inequality has as its set of solutions the interval.Inclusive inequalities with the “or equal to” component are indicated with a closed dot on the number line and with a square bracket using interval notation.
Strict inequalities without the “or equal to” component are indicated with an open dot on the number line and a parenthesis using interval notation. Solve compound inequalities—OR Solve absolute value inequalities Sometimes it helps to draw the graph first before writing the solution using interval notation.
Remember to apply the properties of inequality when you are solving compound inequalities. The next example involves dividing by a negative to isolate a variable.
Absolute value inequalities require a slightly different approach. You can rewrite the inequality in double inequality form and solve appropriately when the inequality is “less than.” Below is an example. You use the “or” concept to solve both inequalities. The interval notation is as follows.
Interval notation is a simplified form of writing the solution to an inequality or system of inequalities, using the bracket and parenthesis symbols in lieu of the inequality symbols. Intervals with parentheses are called open intervals, meaning the variable cannot have the value of the endpoints.
Watch video · That's my number line. I have negative I have negative So the solution is, I can either be greater than 29, not greater than or equal to, so greater than 29, that is that right there, or I could be less than negative So any of those are going to satisfy this absolute value inequality.
Here are the steps required for Solving Polynomial Inequalities: Step 1: Write the polynomial in the correct form. The polynomial must be written in descending order and must be less than, greater than, less than or equal to, or greater than or equal to zero.
|Solving Inequalities||For details on it including licensingclick here. This book is licensed under a Creative Commons by-nc-sa 3.|
|Section 5 - Solve Absolute Value Inequalities||Usually, this is used to describe a certain span or group of spans of numbers along a axis, such as an x-axis. However, this notation can be used to describe any group of numbers.|
Step 6: Use interval notation .