Problem 35 - Algebra for College Students

Question

Solve each inequality. Graph the solution set, and write it using interval notation. $$ |5 x+2|>10 $$

Step-by-Step Solution

Answer

In interval notation, the solution is \( (3, \infty ) \).

Step 1: Distribute the negative sign

First, distribute the negative sign through the expression inside the parentheses: \(-(4+r)+2-3r<-14\) will become: \(-4 - r + 2 - 3r <-14\)

Step 2: Combine like terms

Combine the like terms on the left side of the inequality: \(-4 - r + 2 - 3r\) becomes \(-4 + 2 - r - 3r\), which simplifies to: \(-2 - 4r <-14\)

Step 3: Isolate the variable term

Add 2 to both sides to isolate the variable term: \(-2 - 4r + 2 <-14 + 2\) This simplifies to: \(-4r <-12\)

Step 4: Solve for the variable

Divide both sides by -4, and remember to reverse the inequality sign when dividing by a negative number: \(\frac{-4r}{-4} \frac{-12}{-4}\) This simplifies to: \(r > 3\)

Step 5: Write the solution in interval notation

The inequality \(r > 3\) in interval notation is written as: \( (3, \infty ) \)

Step 6: Graph the solution set

To graph the solution \(r > 3\), draw a number line and place an open circle at 3. Shade the line to the right of 3 to indicate that all numbers greater than 3 are included in the solution set.

Key Concepts

Solving Inequalities

When solving inequalities, the goal is to find all possible values of the variable that make the inequality true. It's similar to solving equations, but with a special rule: if you multiply or divide both sides of the inequality by a negative number, you must reverse the inequality sign. In the exercise, we started with \(-(4+r)+2-3r<-14\). By carefully distributing and combining like terms, we isolated the variable on one side to find the solution. Remember to always check each step to make sure you are simplifying correctly.

Here are the steps we followed:

• Distribute negative signs
• Combine like terms
• Isolate the variable term
• Divide and reverse the inequality sign if necessary

This method ensures we find all solutions that satisfy the inequality.

Interval Notation

Interval notation is a way to describe the set of solutions to an inequality. It uses parentheses and brackets to show which numbers are included or excluded. For example, in the inequality \( r > 3 \), only numbers greater than 3 are solutions, not 3 itself. So, we use an open parenthesis: \ (3, \infty ) \.

There are a few key symbols in interval notation:

• Round brackets ( ) : Means the number is not included. Used for 'greater than' (>) or 'less than' (<). Example: (3, ∞)
• Square brackets [ ] : Means the number is included. Used for 'greater than or equal to' (≥) or 'less than or equal to' (≤). Example: [3, ∞)
• Infinity (∞) : Always uses round brackets because infinity is not a number that can be included. Example: (3, ∞)

Always carefully write the solution set in interval notation after solving the inequality.

Graphing Solutions

Graphing solutions of inequalities visually shows which numbers satisfy the inequality. We use a number line to represent this. For example, to graph \( r > 3 \), you place an open circle on 3 (showing that 3 is not included) and shade the line to the right of 3 to indicate that all greater numbers are solutions.

To graph an inequality:

• Draw a number line.
• Mark the point where the variable's value is compared.
• Use an open circle for 'greater than' (>) or 'less than' (<).
• Use a closed circle for 'greater than or equal to' (≥) or 'less than or equal to' (≤).
• Shade to the right for 'greater than' and to the left for 'less than' to cover all possible solutions.

Graphing helps in visualizing the solution set clearly and verifying the correct values included in the solution.