3.1 Linear Inequalities; Absolute Value

Key thing to remember when graphing inequalities. First thing is that if they have one variable, it is to be graphed on a number line. If there are two variables it is to be graphed in the Cartesian Plane. Second thing to remember if it is graphed on a number line and the inequality includes the ≤ or ≥ signs then you are to use a closed dot. If the inequality uses the signs then you are to use a open dot. The key concept about less than or greater than signs is the idea that we can approach a certain value with the understanding that we can get extremely close to that value (as close as we want) but we can never get exactly to that value. The ≥ or ≤ signs present the idea that we can get exactly to that value.

Simply solve the inequality as if you were solving a regular equation. The important thing to remember is that when you divide or multiply both sides by a negative number, then the inequality sign needs to flip directions. Graph the following inequalities. Basic linear inequalities

Remember that absolute value is the distance an point is away from zero, and distance is always a positive number. Therefore if we were going to graph |x|=4, we would plot the points of -4 and 4 on the number line because they are both 4 units away from zero. So the rule for graphing |x|=c, you graph x=c and x=-c Absolute Value

What happens when |x|<c, this is saying that the distance from x t0 0 is less than c units. So the rule for graphing |x|<c, you graph –c<x<c

When |x|>c, this is saying that the distance from x to 0 is more than c units. So the rule for graphing |x|>c comes in two parts, graph x>c and x<-c

You must apply this rule in order to do any math, you cannot do any operations to the terms that are locked inside of the inequality bars Solving sentences involving inequalities. Other than the actual coefficients, what is different between the first inequality and the second inequality? YOU MUST ADDRESS THIS FIRST BEFORE APPLYING the above rule

The rule for solving this algebraically is as follows: Now you are left with an equation with 3 parts, still solve for x, but what you do the part dealing with x, you must do the same thing to all 3 parts. What if |ax+b|<c

Final possibility First inequality you simply drop the absolute value bars and solve the inequality like it were an equation. For the second part you drop the absolute value bars, flip the inequality sign, and change the sign of your constant term “c” Solve both new inequalities and graph them on the same number line.

What is the solution for the following Quick question