Comparing Methods for Solving Linear Equations and Inequalities – Commonalities, Differences, and the Concept of Functions

Comparing Methods for Solving Linear Equations and Inequalities – Commonalities, Differences, and the Concept of Functions

Question 1a

Linear equations and inequalities are not much different. The methods for solving them are also quite similar, as the expression is adjusted to achieve the variable on one side of the equation. Thus, to solve equations and/or inequalities, we perform operations on both sides of the equation/inequality to preserve equality. For instance, let’s assume that we have the linear equation 3x + 5 = 17. To solve for x, we would subtract 5 from both sides to get 3x = 12 and divide both sides by 3 to get x= 4. Similarly, for the linear inequality 3x + 5 ≤ 17, we would follow the same steps: So by subtracting 5 from both sides, we get 3x ≤ 12, and by dividing both sides by 3, we get x ≤ 4.

The major difference is rooted in how to operate multiplication or division with a negative number. In linear equations, operations of addition and subtraction do not affect the signs of the numbers, but multiplication or division by a negative number does not introduce a new inequality. For example, in -2x = 6, two operations that are allowed are the cancellation of -2 on the left-hand side, giving -2x/ -2 = x => x = -3. However, specific rules that pertain to linear inequalities are that when multiplying or dividing by a negative number, the direction of the inequality sign is changed. If in the case of -2x ≥ 6 to divide by -2, we get 2x ≥ -3, we must then reverse the sign to get -2x ≤ -3. This difference arises from the nature of inequalities: we lose the position of smaller and greater when multiplying an inequality by a negative number, and that is the reason for the need for a sign flip.

Question 2a

Relation in mathematics is a rather general idea of a link between the elements of two sets. For instance, the operation “is greater than” for real numbers or “is the sister of” for people. Nevertheless, not all relations can be referred to as functions. A function is a special type of relation with a specific property: every input in the domain (all the possible inputs should result in exactly one possible output in the codomain.

I appreciated this distinction when expressing the relations and functions through the use of arrow diagrams or graphs. With regards to an arrow diagram for a relation, it is possible to have one or several arrows that point from an element in the domain to more than one element in the codomain. For example, one can find the elements one and three in both sets {1, 2, 3, 4} and {1, 3, 5, 7} at the same time because one is a factor of zero and three is a factor of zero, respectively. This is a valid relation, but it does not qualify as a function because one domain (2) can produce more than one range (4 and 8).

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Think and Answer requires you to answer two questions. For each question, do not simply provide an answer; make sure you explain how you arrived at that answer. Even if your reasoning is wrong, you will still be credited for participation.

1a. Compare methods of solving linear equations and methods of solving linear inequalities. What do they have in common? What is different?

Comparing Methods for Solving Linear Equations and Inequalities – Commonalities, Differences, and the Concept of Functions

2a. Why not all relations can be called functions?