Problem 54 - Beginning and Intermediate Algebra

Question

The following exercises contain absolute value equations, linear inequalities, and both types of absolute value inequalities. Solve each. Write the solution set for equations in set notation and use interval notation for inequalities. $$|2 p-5|-12=11$$

Step-by-Step Solution

Answer

The short answer for the given absolute value equation is: \(p = 9\) or \(p = 14\), and the solution set is \(\{9, 14\}\).

Step 1: Isolate the absolute value expression

We are given the absolute value equation \(|2 p-5|-12=11\). To isolate the absolute value expression, we need to add 12 on both sides of the equation. $$ |2 p-5|=11+12 $$ Simplifying, we obtain: $$ |2 p-5|=23 $$

Step 2: Create two linear equations

Now that the absolute value expression is isolated, we will create two separate linear equations: one for when the expression inside the absolute value is positive, and another for when the expression is negative. If \(2p-5\) is positive or equal to zero, we have: $$ 2p-5 = 23 $$ If \(2p-5\) is negative, we have: $$ -(2p-5) = 23 $$

Step 3: Solve each linear equation

Now we need to solve each linear equation obtained in step 2: For the first equation, \(2p-5 = 23\), we will add 5 to both sides and then divide by 2: $$ 2p = 28 \Rightarrow p = 14 $$ For the second equation, \(-(2p-5) = 23\), we will first distribute the negative sign: $$ -2p+5 = 23 $$ Now we will subtract 5 from both sides and then divide by -2: $$ -2p = -18\Rightarrow p = 9 $$

Step 4: Write the solution set in set notation

Since we have found the two possible values for p, we can write the solution set in set notation, which is: $$ \{9, 14\} $$ So the solution to the given absolute value equation is \(p = 9\) or \(p = 14\), and the solution set is \(\{9, 14\}\).

Key Concepts

Linear Inequalities

Linear inequalities are mathematical expressions that involve inequality symbols like \

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• \(<\)
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• \(>\)
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• \(\leq\)
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• \(\geq\)
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\These symbols help us compare expressions and determine the range of possible solutions. A typical linear inequality can be written as \(ax + b < c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. To solve linear inequalities, follow these steps:\

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• First, treat the inequality like a linear equation. Perform basic operations to simplify and isolate \(x\) on one side of the inequality sign.
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• If you multiply or divide by a negative number, remember to flip the inequality sign.
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• The solution can be expressed in inequality form, interval notation, or graphically on a number line.
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\Understanding linear inequalities is fundamental to solving more complex equations, such as absolute value inequalities.

Absolute Value Inequalities

Absolute value inequalities involve expressions within absolute value symbols, like \

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• \(|x| \leq a\)
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• \(|x| \geq a\)
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\These express conditions where the distance of a variable from zero is constrained by a limit \(a\). There are two main approaches to solving absolute value inequalities depending on the inequality sign:\

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• If the form is \(|x| < a\), the solution breaks into two linear inequalities: \(-a < x < a\).
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• If the form is \(|x| > a\), the solution becomes two separate conditions: \(x < -a\) or \(x > a\).
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\Always remember that absolute values signify distance, so the solutions focus on the range of values for \(x\) that fit within or beyond a certain threshold.

Interval Notation

Interval notation is a concise way of representing a set of numbers, particularly for the solution sets of inequalities. It uses brackets and parentheses to indicate whether endpoints are included or excluded:\

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• Square brackets \([ ]\) indicate that the endpoint is included, known as "closed interval."
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• Parentheses \(( )\) indicate that the endpoint is not included, known as "open interval."
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\For example, the inequality \(2 < x \leq 5\) is expressed in interval notation as \((2, 5]\). Here, 2 is not included, while 5 is. Interval notation simplifies describing continuous sets of numbers, making it a very handy tool in mathematics, especially when working with inequalities.

Set Notation

Set notation is a method to express a collection of elements, often used to specify the solutions to equations or systems in a clear, unambiguous way. In mathematics, you denote a set by enclosing its elements with curly braces \(\{ \}\).\For example, if you solve an equation and find solutions 9 and 14, you can state the solution set as \(\{9, 14\}\).\This notation is finite, listing explicitly every distinct solution, unlike interval notation that often needs brackets or signs to express an infinite set. Set notation is especially useful for situations where solutions are discrete and not continuous, which often occurs in equations involving absolute values.