Problem 77 - 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. $$|5 j+3|+1 \leq 9$$

Step-by-Step Solution

Answer

The solution to the inequality \(|5j + 3| \leq 8\) is the interval \(\left[-\dfrac{11}{5}, 1\right]\).

Step 1: Isolate the absolute value expression

First, let's isolate the absolute value expression by subtracting 1 from both sides of the inequality: $$|5j + 3| \leq 8$$

Step 2: Split the inequality into two cases

Now, let's split the inequality into two cases: one case for the positive value of the expression inside the absolute value and another case for its negative value. Case 1 (positive value): $$5j + 3 \leq 8$$ Case 2 (negative value): $$(5j+3) \geq -8$$

Step 3: Solve each inequality

We will now solve each inequality separately. Case 1: \begin{align*} 5j + 3 &\leq 8 \\ 5j &\leq 5 \\ j &\leq 1 \end{align*} Case 2: \begin{align*} (5j+3) &\geq -8 \\ 5j &\geq -11 \\ j &\geq -\dfrac{11}{5} \end{align*}

Step 4: Write the final solution in interval notation

The solution to the inequality \(|5j + 3| \leq 8\) is the intersection of both cases. In interval notation, this is represented as: $$\left[-\dfrac{11}{5}, 1\right]$$

Key Concepts

Linear Inequalities

Linear inequalities are expressions that, unlike linear equations, involve inequality signs rather than equal signs. For example, symbols like \( <, >, \leq, \geq \) are used. These symbols help us understand the relationship between two expressions, showing whether one is smaller, greater, or even equal and smaller/greater at the same time.
To solve a linear inequality, you follow similar procedures as those with linear equations. Here's what you typically do:

• Isolate the variable on one side of the inequality.
• Add or subtract terms on both sides to simplify the inequality.
• Divide or multiply both sides of the inequality by a constant to solve for the variable.

One key rule to remember is that when multiplying or dividing by a negative number, the inequality sign must be flipped. For instance, if \( -3x > 6 \), dividing both sides by \(-3\) reverses the inequality, so \( x < -2 \).
Understanding linear inequalities is crucial because they lay the groundwork for working with more complex expressions like those involving absolute values.

Interval Notation

Interval notation provides a convenient way to express solutions to inequalities. Instead of listing numerous values, you can express a range of values compactly. This saves time and avoids errors. Interval notation uses brackets and parentheses to denote included and excluded endpoints, respectively.
Here's how it works:

• Inclusive endpoints: Use square brackets \([ ]\) when the endpoint belongs to the range. For example, \([1, 5]\) means "all numbers from 1 to 5, including those endpoints."
• Exclusive endpoints: Use parentheses \(( )\) when the endpoint does not belong to the range. For example, \((1, 5)\) indicates "all numbers between 1 and 5, not including the endpoints."

Combined, such as in \([-\frac{11}{5}, 1]\), indicates a range from \(-\frac{11}{5}\) to \(1\) where both \(-\frac{11}{5}\) and \(1\) are included. This notation streamlines expressing the result, especially for complex inequalities like those involving absolute values.

Absolute Value Equations

Absolute value equations involve expressions within an absolute value symbol, \(|\cdot|\), which represents the distance from zero on the number line, making it always non-negative. This feature means that solving these equations often involves considering multiple cases.
Typically, an equation with an absolute value \(|ax + b| = c\) can result in two different equations:

• Case 1: \( ax + b = c \) (solving the expression inside the absolute normally)
• Case 2: \( ax + b = -c \) (considering the negative counterpart)

These two cases arise because both \(c\) and \(-c\) have the same absolute value. The solution depends on solving both linear discussions and conjoining their solutions. For example, \(|5j + 3| \leq 8\) requires checking both "\(+\)" and "\(-\)" cases separately and merging results as needed.
Understanding absolute value equations is essential as they frequently occur in math problems and everyday contexts, like determining error margins or deviations.