Homework this week: 1.5b, 1.7ab Quiz 2: 1.4c, 1.5b, 1.7ab 1. Solving Formulas A formula is an equation which represents the relationship between two or more variables. Example 1.1. (1) The formula x = y means that x and y are the same. (2) The formula C = 5 (F − 32) represents the relationship between temperatures in 9 degrees Fahrenheit F and Celsius C. A formula is called solved for one of its variables if that variable is isolated on one side of the equals sign in the equation. Not all formulas can be solved for all of their variables. Example 1.2. The formula we wrote relating Fahrenheit and Celsius temperatures is solved for the variable representing degrees in Celsius C. We solve it for degrees in Fahrenheit F : 5 C = (F − 32) 9 9 C = F − 32 5 9 C + 32 = F 5 Written in the normal way (with the solved variable on the left-hand side) our solution is 9 F = C + 32 5 2. Solving Linear Inequalities 2.1. Inequalities and Interval Notation. An inequality is an expression representing the way that variables or numbers are not equal. We will learn to simplify expressions which describe how a variable is less or greater than another. In mathematical notation for fixed real numbers a, the inequality x < a represents all numbers less than a; we can write this equivalently a > x. The inequality x > a represents all numbers greater than a. The previous inequalities are called strict, since we are not allowed to take x = a. If we can take x = a, the inequalities are written x ≤ a or x ≥ a. The solution set of an inequality can be written in set-builder notation: for example, the real numbers greater than or equal to a can be written as {x : x > a} When the upper bound on the solution set is missing, we can write it with +∞; similarly, if the lower bound is missing, we can write −∞. That is, {x : x > a} = {x : +∞ > x > a}
Date: Spring 2014.
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In this way, every inequality can be written with two bounds. We are also interested in this type of compound inequalities, which take the form, for instance, of a 0, then ax < ay. If a < 0, then ax > ay. The same rules apply in the case of non-strict inequalities, in an obvious way. Using these rules, we can solve an inequality just like we would solve a formula. Exercise 2.1. Solve the following inequalities, writing the solution in interval notation (1) 2x + 5 > −3 (2) −4x ≤ 10 (3) 2 < x − 1.1 < 20 (4) −1 ≥ x + 4 > −8

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3. Solving Absolute Value Inequalities Here, we will learn to solve absolute value inequalities; that is, inequalities of the form |ax + b| < c |ax + b| ≤ c |ax + b| > c and |ax + b| ≥ c For c > 0. When we have written the absolute value inequality in this form, we say it is in standard form. Standard form is useful because we can immediately tell whether the absolute value inequality has a solution. There is first a special case to be aware of: when c < 0 the situation will change. However, in each case it changes in what seems very intuitive, based on what we know about absolute values in general. If c < 0, then the equations |ax + b| < c and |ax + b| ≤ c have no solution, since in this case they both state |ax + b| < 0, which is impossible. On the other hand, if c < 0, then the equations |ax + b| > c and |ax + b| ≥ c do have a solution, and in fact the solution is all real numbers, which we can write in interval notation as (−∞, ∞). More on interval notation in a bit. Our technique to solve an absolute value equation with the absolute value on the left with either < or ≤ is similar: assume we have strict inequality 0, we write the equation |ax + b| < c As an equivalent, compound inequality, −c < ax + b < c Now, so long as we are careful to make any change to one term also to the other two, and follow the rules regarding inequalities, we will derive an inequality for x by itself; assuming that a > 0 we would have, for instance −c − b c−b x has no lowest value which makes it true. In interval notation, we always need a lower and upper point, so it is natural to replace these boundary points with −∞ or ∞, depending on if we need a lower value or upper value. These values are never included with square brace, since they are not real numbers. So, an inequality of the form x < b can be written in interval notation as (−∞, b). Similarly, the inequality x ≤ b can be written (−∞, b] (4) The inequality x > a > can be written in interval notation as (a, ∞), and the inequality x ≥ a can be written in interval notation as [a, ∞). To combine intervals, we may use many tools. The two which we use in this section are the union and intersection. The union of two intervals (a, b) and (c, d), denoted by (a, b) ∪ (c, d) is the collection of points which are in either (a, b) or (c, d). The intersection of two intervals (a, b) and (c, d), denoted by (a, b) ∩ (c, d) is the collection of points which are in both intervals. Note 3.1 (Compound Inequalities). It is important to understand how the compound inequality of the form a < x < b is defined (we have used it multiple times already without this understanding, perhaps.) Specifically, we write a < x < b if and only if a < x andx < b Other compound inequalities are defined similarly. When working with a compound inequality, which generally makes problem solving faster, if being careless one can arrive at a nonsensical statement. For instance, if one finds −1 < x ≤ −5 The solution set corresponding to this inequality is empty, since there are no numbers greater than one and less than −5! If we expected a solution, we should check our work. Let us now continue with our absolute value inequality solution techniques. The other case which may come up is when the absolute value is to the left of a > or ≥ sign. In this case, we follow the following rule: If |ax + b| > c for some c > 0, then ax + b > c or −(ax + b) > c is true. It is in this case that we will oftentimes require the use of the union symbol ∪. Example 3.1. Let us solve some problems of this type; write the solution in interval notation (1) |x − 5| > 6 1 (2) 4x−3 > 2 2 (3) |x + 6| < − 1 | 2 (4) |2x + 4| + 4 < 7 (5) 7 > |8 − 5x| + 3 (6) −4 < |6x − 1| − 4

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E-mail address: jgoldfar@my.fit.edu Mathematical Sciences Department, Florida Institute of Technology, 150 W. University Blvd., Melbourne, FL 32901