Problem 68 - 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. $$4+3(2 r-5)>9-4 r$$

Step-by-Step Solution

Answer

The solution to the given inequality is \(r > 2\), expressed in interval notation as \((2, \infty)\).

Step 1: Distribute the term inside the parentheses

In this step, we will apply the distributive property to simplify the inequality. $$4+3(2 r-5)>9-4 r$$ $$4 + 6r - 15 > 9 - 4r$$

Step 2: Combine like terms

Combine the constants and the terms with variable $$r$$ on both sides of the inequality: $$6r - 11 > -4r + 9$$

Step 3: Isolate the variable on one side

To isolate $$r$$, add $$4r$$ to both sides and add $$11$$ on both sides: $$6r - 11 + 4r + 11 > -4r + 9 + 4r + 11$$ $$10r > 20$$

Step 4: Solve for the variable

To solve the inequality for $$r$$, divide both sides by 10: $$\frac{10r}{10} > \frac{20}{10}$$ $$r > 2$$

Step 5: Write the solution in interval notation

Since the inequality is $$r > 2$$, the interval notation for the solution set is: $$ (2, \infty)$$ So the solution to the given inequality is $$r > 2$$, expressed in interval notation as $$(2, \infty)$$

Key Concepts

Absolute Value Equations

Absolute value equations involve equations where the variable is inside an absolute value expression, such as \(|x| = a\). To solve these, we need to consider both the positive and negative scenarios that could result in the given absolute outcome.

• Start by isolating the absolute value on one side of the equation. If your absolute value equation looks like \(|Ax + B| = C\), make sure \(C\) is positive since the absolute value cannot be negative.
• Split the equation into two separate cases: one where the expression inside the absolute value is equal to \(C\) and another where it equals \(-C\).
• Solve each equation separately to find all possible solutions.

Remember, absolute value functions indicate the distance from zero on a number line, and each scenario reflects that distance both in the positive and negative directions.

Linear Inequalities

Linear inequalities, like equations, involve algebraic expressions with a variable. The critical difference is that inequalities use symbols like \(>\), \(<\), \(\geq\), and \(\leq\) to express a range of values rather than a specific one.

• Much like linear equations, simplifying, distributing, and isolating terms are essential.
• When you multiply or divide by a negative number, remember to flip the inequality sign.

For example, in a problem like \(4 + 6r - 15 > 9 - 4r\), you first simplify by combining like terms and then isolate \(r\) to find its range (like \(r > 2\)). Always verify your result by substituting numbers within and outside your determined range to check your understanding.

Set Notation

Set notation helps express the solution to an equation clearly and systematically. When you solve equations (not inequalities), you often use set notation to list all possible solutions. Let's see some common elements:

• A solution set is a collection of answers that satisfy the equation. For instance, if you solved \(x^2 = 4\) and found \(x = 2\) and \(x = -2\), you would write the solution set as \(\{2, -2\}\).
• Curly braces \(\{ \}\) are used in set notation to list distinct solution values.

Set notation is more applicable for distinct and discrete values, often used in problems involving absolute value equations.

Interval Notation

Interval notation is a convenient way to express solutions of inequalities, indicating all the numbers between a set of boundaries or endpoints.

• If the inequality is strict (\(<\) or \(>\)), use parentheses \((\) to show that the boundary is not included.
• If the inequality includes the boundary (\(\leq\) or \(\geq\)), use square brackets \([ \) instead.
• For example, the inequality \(r > 2\) is represented as \((2, \infty)\).
• \(\infty\) and \(-\infty\) always have parentheses because they indicate unbounded ranges.

Interval notation simplifies expressing continuous solutions that arise in inequalities.