Problem 56 - 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. $$\frac{1}{2} n+11<8$$

Step-by-Step Solution

Answer

The solution to the inequality \(\frac{1}{2}n + 11 < 8\) is \(n < -6\), expressed in interval notation as \((-\infty, -6)\).

Step 1: Write down the inequality

Given the inequality \(\frac{1}{2}n + 11 < 8\)

Step 2: Isolate n

Subtract 11 from both sides of the inequality. \(\frac{1}{2}n + 11 - 11 < 8 - 11\) which simplifies to: \(\frac{1}{2}n < -3\)

Step 3: Solve for n

Multiply both sides by 2 to eliminate the fraction. \((\frac{1}{2}n) \times 2 < -3 \times 2\) This gives us: \(n < -6\)

Step 4: Write the solution in interval notation

The solution for the inequality \(n < -6\) can be expressed as the following interval: \((-\infty, -6)\)

Key Concepts

Absolute Value Equations

Absolute value equations involve expressions where the absolute value of a variable or expression is set equal to a number. The absolute value of a number is its distance from zero on the number line, disregarding the sign. For example, in the equation \(|x| = 3\), \(x\) can be either 3 or -3, because both numbers have an absolute value of 3.
To solve absolute value equations, you need to consider both the positive and negative scenarios of the expression inside the absolute value:

• Set the inside expression equal to the positive value.
• Set the inside expression equal to the negative value.

This way, you find all possible solutions for \(x\). Just remember, the absolute value equation \(|x| = a\) only has solutions if \(a\) is non-negative.

Absolute Value Inequalities

Absolute value inequalities express a range of values in terms of the absolute value of an expression being greater than or less than a number. For example, \(|x| < 5\) indicates that \(x\) can be any number from -5 to 5. This is because the absolute value must be less than 5.
To solve absolute value inequalities, consider these rules:

• For \(|x| < a\), write it as \(-a < x < a\).
• For \(|x| > a\), write it as \(x < -a\) or \(x > a\).

These rules help you find the range of values that satisfy the inequality, providing a clear visual of all the possible solutions.

Set Notation

Set notation is a standard way of expressing a collection of objects or numbers. When you solve an equation and want to express the solution clearly, set notation comes in handy. For instance, if you solved \(x = 3\), the solution in set notation would be \(\{3\}\).
Here are some basics:

• Use curly braces \(\{\}\) to define the set.
• List each element explicitly, separated by commas.
• For more complex sets, use descriptions like \(\{x | x > 0\}\), meaning all \(x\) greater than 0.

Set notation is concise and clean, making it easier to identify the solutions provided by solving equations.

Interval Notation

Interval notation is a concise way to describe ranges of values, often used for solutions to inequalities. Unlike set notation, interval notation doesn't list individual elements but instead provides the range.
Here’s how interval notation works:

• Use parentheses \(( )\) for numbers not included in the set, and brackets \([ ]\) for numbers that are included.
• For example, \((a, b)\) means all numbers between \(a\) and \(b\), but not including \(a\) or \(b\).
• \([a, b]\) includes both boundary numbers \(a\) and \(b\).
• Use \(-\infty\) or \(+\infty\) to indicate no end point in one direction, always in parentheses, such as \((-\infty, a]\).

Interval notation is straightforward and helps in graphing the solution set on a number line quickly.