Problem 69 - 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. $$|6 k+17|>-4$$

Step-by-Step Solution

Answer

The solution to the inequality \(|6 k+17| > -4\) is \(k \in \left( -\frac{7}{2}, -\frac{13}{6} \right)\).

Step 1: Understand the definition of absolute value

The absolute value of a number x is denoted as |x| and is defined as the distance of x from 0 on the number line. It is always positive or zero, never negative. The mathematical definition of the absolute value is: |x| = { x if x ≥ 0 -x if x < 0 Now, we apply this definition to the given inequality.

Step 2: Break the absolute value inequality into two separate inequalities

Given an inequality of the form: $$|x| > a$$ This can be rewritten as two linear inequalities combined by AND: $$x < -a \text{ AND } x > a$$ For the given inequality, \(|6 k+17| > -4\), we will break it into two separate inequalities.

Step 3: Write the inequality without absolute value

Rewrite the inequality \(|6 k+17| > -4\) as two linear inequalities: $$(6k+17) < 4 \text{ AND } (6k+17) > -4$$

Step 4: Solve the inequalities

Now, we will solve both inequalities and get the solution for k. For the first inequality: $$6k + 17 < 4$$ Subtract 17 from both sides: $$6k < -13$$ Divide both sides by 6: $$k < -\frac{13}{6}$$ For the second inequality: $$6k + 17 > -4$$ Subtract 17 from both sides: $$6k > -21$$ Divide both sides by 6: $$k > -\frac{7}{2}$$

Step 5: Write the final answer in interval notation

Now that we have the solutions for k, we will write it in interval notation as: $$k \in \left( -\frac{7}{2}, -\frac{13}{6} \right)$$

Key Concepts

Absolute Value Equations

Absolute value equations involve expressions within absolute value bars, such as \(|x|\). The absolute value refers to a number's distance from zero on the number line, always a non-negative value. This makes it a powerful tool in solving equations, especially when dealing with values where direction (positive or negative) is irrelevant.
For an equation like \(|x| = a\), it translates to two possible equations:

• \(x = a\): This is when the number itself is a units away from 0.
• \(x = -a\): This is when the opposite of the number is also a units away from 0.

This dual nature allows absolute value equations to capture two potential solutions, which is particularly handy in mathematical modeling and real-world applications. Remember, the value 'a' must be non-negative since absolute values cannot be negative.

Linear Inequalities

Linear inequalities share similarities with linear equations, except they incorporate inequality symbols such as \(<\), \(>\), \(\leq\), or \(\geq\). These inequalities represent a range of possible solutions rather than a single point. To solve a linear inequality:

• Treat it like a regular equation; perform operations such as addition, subtraction, multiplication, or division on both sides.
• Important: When multiplying or dividing by a negative number, the inequality sign must be flipped.

After solving a linear inequality, you identify a range of values on the number line, which represents the potential solutions. This approach helps in determining constraints and limits, making it immensely applicable in scenarios like budgeting and comparing quantities.

Interval Notation

Interval notation is a concise way to express a range of numbers, widely used to present solutions of inequalities. Instead of listing all numbers within a range, interval notation uses brackets and parentheses to define the interval:

• Round brackets \(( )\) indicate that the endpoint is not part of the interval (open interval).
• Square brackets \([ ]\) signify that the endpoint is included in the interval (closed interval).

For example, the solution \(k \in \left( -\frac{7}{2}, -\frac{13}{6} \right)\) denotes that all values of \(k\) fall between \(-\frac{7}{2}\) and \(-\frac{13}{6}\) but do not include the endpoints themselves. This concise way of representing ranges is extensively used in calculus and high school algebra.

Set Notation

Set notation is another method to describe a collection of numbers or solutions, often using curly braces \(\{ \}\). This method is useful for representing solutions to absolute value equations, where specific distinct solutions may be found. In set notation, you explicitly list out each solution:

• Utilized for finite, discrete contributions.
• Expresses inclusivity of exact solutions derived from equations.

For example, if an absolute value equation yields solutions \(x = 3\) and \(x = -3\), you would write the solution set as \(\{3, -3\}\). This form is particularly useful when it is necessary to catalog solutions manually, offering a clear identification of specific solution points.