Problem 50 - 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. $$\left|\frac{5}{3} a+2\right|=8$$

Step-by-Step Solution

Answer

The solution set for the given absolute value equation is \(a\in\{-6, \frac{18}{5}\}\).

Step 1: Write down the given equation

Given the absolute value equation: \[\left|\frac{5}{3} a+2\right|=8\]

Step 2: Remove the absolute value sign

To remove the absolute value sign, we will create two separate equations. One will be the same as the equation inside the absolute value, and the other will be the negative of the equation inside the absolute value. This results in two linear equations: \[\frac{5}{3}a + 2 = 8,\] \[\frac{5}{3}a + 2 = -8.\]

Step 3: Solve each linear equation

We will solve the two linear equations separately. For the first equation: \[\frac{5}{3}a + 2 = 8\] Subtract 2 from both sides: \[\frac{5}{3} a = 6\] Multiply both sides by 3/5 to isolate "a": \[a = \frac{18}{5}\] For the second equation: \[\frac{5}{3} a + 2 = -8\] Subtract 2 from both sides: \[\frac{5}{3} a = -10\] Multiply both sides by 3/5 to isolate "a": \[a = -6\]

Step 4: Write the solution set in set notation

Now that we have solved for "a" in both separate equations, we can write the solution set in set notation: \[{a}=\left\{-6, \frac{18}{5}\right\}\] So our final answer is \(a\in\{-6, \frac{18}{5}\}\).

Key Concepts

Understanding Set Notation

Set notation is a method used to define a collection of elements, often numbers in mathematics. This notation clearly lists the possible solutions to an equation or inequality. It's particularly useful when dealing with multiple solutions, as seen in absolute value equations.

Take our original exercise as an example. After solving the absolute value equation, we discovered two solutions: \(a = -6\) and \(a = \frac{18}{5}\).

• To express these solutions using set notation, we list them inside curly braces.
• This set clearly shows all possible values of \(a\) that satisfy the given equation.

So, the solution set is written as \(a \in \{-6, \frac{18}{5}\}\). This notation helps us quickly understand which numbers are included without additional context.

Solving Linear Equations

Linear equations involve expressions that equal a constant value, without exponents higher than one. They form straight lines when graphed, hence the term 'linear.'

In our exercise, once we removed the absolute value, we had two linear equations to solve:

• \(\frac{5}{3}a + 2 = 8\)
• \(\frac{5}{3}a + 2 = -8\)

For each equation, the goal is to isolate the variable \(a\).

**Steps to Solve a Linear Equation**

1. Begin by eliminating any constants from one side of the equation. Here, we subtracted 2 from both sides.
2. After simplifying, isolate the variable \(a\) by performing the inverse operation of what's affecting \(a\). In our example, we multiplied both sides by \(\frac{3}{5}\) to get \(a\) alone.

Solving these linear equations reveals the values of \(a\) as \(-6\) and \(\frac{18}{5}\). Understanding how to systematically isolate and solve for the variable is crucial in algebra.

Exploring Interval Notation

Interval notation is used to represent a range of values, typically for inequalities. Unlike set notation, it captures an entire span of numbers, showing where the solutions lie between.

In the context of inequalities, you might see intervals like \((a, b)\), which include all numbers between \(a\) and \(b\) but not the endpoints. Alternatively, if the endpoints are included, we use square brackets: \([a, b]\).

For example, if we were given an inequality instead of an absolute value equation and found that \(a\) could range from \(-6\) to \(\frac{18}{5}\), we’d use interval notation to show that range accurately. Keep in mind:

• Parentheses \((a, b)\) represent an open interval, meaning \(a\) and \(b\) are not included.
• Brackets \([a, b]\) describe a closed interval, meaning \(a\) and \(b\) are included.

This notation succinctly communicates a continuous range of values, making it useful for expressing solutions to inequalities.