Understanding One-Variable Linear Inequalities | Johns Hopkins University

University:
Johns Hopkins University
Course:
AS.110.601 | Algebra I
Academic year:

2024
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17

Pages:

2
Author:

anon

Understanding One-Variable Linear Inequalities Mathematics' use of linear inequalities in one variable is essential for understanding and resolving problems involving relationships between numbers. Equations that express an inequality are those in which one of the two quantities is either bigger than or less than the other. We will look at the characteristics of linear inequalities in one variable in this post, as well as how to leverage these characteristics to solve issues. The Basics of Inequalities An inequality is, in essence, defined as when two real numbers, a and b, are such that b minus an is a positive real number. If this is the case, we say that an is less than b and express this as a b. A is to the left of b on the number line, and there is some space between the two figures. Inferring the converse from this statement, we can state that b is greater than a by writing 𝑏 > 𝑎. When comparing two real numbers, there are three possible results: either an is less than b, b is less than a, or an is equal to b. It's vital to keep in mind that these are the only three possibilities that can occur due to the law of trichotomy. Properties of Inequalities When dealing with linear inequalities in a single variable, we need to be mindful of four characteristics of inequalities. These attributes are: 1. Adding c to both sides of an inequality does not modify the inequality's sign if an is less than b and c is a real number. The indication does not change. 2. Multiplying both sides of the inequality by c will still result in a being less than b if an is less than b and c is a positive real number. 3. If both sides of the inequality are multiplied by c and an is smaller than b and c is negative, a will be higher than b. The only algebraic issue to be aware of when dealing with inequalities is this one. 4. The inequality's sign must be changed if both sides are divided by a negative number. Solving Linear Inequalities in One Variable With one significant exception, solving linear inequalities in one variable is similar to solving equalities. We must exercise caution when multiplying or dividing by negative values. The same steps are used to solve an inequality as they are for an equality, but we must bear in mind the aforementioned characteristics of inequalities. Let's use an example to demonstrate this. Take into account the inequality 2𝑥 − 1 < 4𝑥 + 3. We approach this inequality just like we would an equality in order to find a solution. In order to eliminate the -1 and isolate x on one side of the inequality, we first add 1 to both sides of the inequality: 2𝑥 < 4𝑥 + 4 Then, to isolate 4x on one side, we subtract 2x from both sides: 0 < 2𝑥 + 4 Finally, to determine the answer, we deduct 4 from both sides. − 4 < 2𝑥 Now that we have divided both sides by 2, we can determine the answer: − 2< 𝑥 The answer to the inequality is x -2, hence x can be any number that is less than -2. Conclusion For the purpose of solving real-world issues, it is crucial to comprehend the properties of and know how to solve linear inequalities in one variable, which are a fundamental mathematical notion. We may confidently and easily answer linear inequalities in one variable by recalling the four characteristics of inequalities.

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