Linear Equations, Inequalities, and Functions

Linear Equations and Inequalities in Two Variables
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All courses. Algebra 1 Discovering expressions, equations and functions Overview Expressions and variables Operations in the right order Composing expressions Composing equations and inequalities Representing functions as rules and graphs. Algebra 1 Exploring real numbers Overview Integers and rational numbers Calculating with real numbers The Distributive property Square roots.

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The basic rule throughout is that whatever you do to one side of the equation you must also do to the other. A linear inequality resembles in form an equation, but with the equal sign replaced by an inequality symbol.

Linear Inequalities

The solution of a linear inequality is generally a range of values, rather than one specific value. Such inequalities arise naturally in problems involving words such as 'at least' or 'at most'.

Linear Functions

To solve an inequality we use the same procedures as we used when solving linear equations, with the modification that when an inequality is multiplied or divided by a negative number, the inequality is reversed. Inequalities also arise when we examine the domain of certain functions.

Introduction to graphing inequalities - Two-variable linear inequalities - Algebra I - Khan Academy

Hence it is important that students are familiar with these before studying functions. This means that our solutions will, in most cases, be inequalities themselves. Solving single linear inequalities follow pretty much the same process for solving linear equations. Now, with this inequality we ended up with the variable on the right side when it more traditionally on the left side.

So, here is the inequality notation for the inequality. The process here is similar in some ways to solving single inequalities and yet very different in other ways.

Solving linear inequalities

The process here is fairly similar to the process for single inequalities, but we will first need to be careful in a couple of places. Our first step in this case will be to clear any parenthesis in the middle term. The only thing that we need to remember here is that if we do something to middle term we need to do the same thing to BOTH of the out terms.

One of the more common mistakes at this point is to add something, for example, to the middle and only add it to one of the two sides.

That is the inequality form of the answer. In this case the first thing that we need to do is clear fractions out by multiplying all three parts by 2.

Standards in this domain:

We will then proceed as we did in the first part. In this step we need to divide all three parts by However, recall that whenever we divide both sides of an inequality by a negative number we need to switch the direction of the inequality. For us, this means that both of the inequalities will need to switch direction here. So, there is the inequality form of the solution.

We will need to be careful with the interval notation for the solution.

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Related Standards F. The solution set of a system of linear inequalities consists of all points whose coordinates satisfy each inequality in the system. This technique worked because we had y alone on one side of the inequality. Naturally, I want below the line. Learn to recognize constant additive change — and, thereby, linear functions - in verbal, tabular and graphical forms of a function. Lesson 5 Absolute Value Functions and Graphs.

Remember that in interval notation the smaller number must always go on the left side! Note as well that this does match up with the inequality form of the solution as well. Here is that form,. Either of the inequalities in the second row will work for the solution. When solving double inequalities make sure to pay attention to the inequalities that are in the original problem.

In other words, it is easy to all of a sudden make both of the inequalities the same.