Problem 34 - Algebra and Trigonometry

Question

In all exercises other than \(\varnothing\), use interval notation to express solution sets and graph each solution set on a number line. In Exercises \(27-50,\) solve each linear inequality. $$ 18 x+45 \leq 12 x-8 $$

Step-by-Step Solution

Answer

The solution to the inequality is \(x \leq -\frac{53}{6}\). In interval notation, this is \((-\infty,-\frac{53}{6}]\).

Step 1: Simplify the inequality

Collect like terms on both sides of the inequality. To do this, subtract \(12x\) from both sides to get \(6x + 45 \leq -8\).

Step 2: Isolate the variable x

Subtract \(45\) from both sides of the inequality to isolate \(x\): \(6x \leq -53\).

Step 3: Solve for x

Divide both sides by \(6\): \(x \leq -\frac{53}{6}\). This is the solution in inequality form.

Step 4: Convert to interval notation & graph the solution

This inequality represents all values \(x\) such that \(x\) is less than or equal to \(-\frac{53}{6}\). In interval notation, this is written as \((-\infty,-\frac{53}{6}]\). Represent this on the number line with a closed dot at \(-\frac{53}{6}\) and an arrow pointing to the left, indicating all numbers less than or equal to \(-\frac{53}{6}\).

Key Concepts

Understanding Interval Notation

Interval notation is a simple and concise way to describe a range of numbers in mathematics. It's especially useful when dealing with inequalities.

• Around the numbers, parentheses \( ( ) \) indicate that the end values are not included in the set. These are called open intervals.
• Brackets \( [ ] \) mean the end values are included, known as closed intervals.

For example, if you have an inequality like \( x \leq -\frac{53}{6} \,\), the corresponding interval notation would be \( (-\infty, -\frac{53}{6}] \). Here, \( -\infty \) always uses a parenthesis because infinity isn't a number we can "reach."
This notation helps to clearly show whether certain endpoints are included or excluded, making it easier to understand and communicate solutions for any inequality.

Solving Linear Inequalities

Solving linear inequalities is much like solving linear equations, but with a few crucial differences. When working with inequalities:

• Start by simplifying each side, just like you would with an equation. For our example, subtract \(12x\) from both sides to get \(6x + 45 \leq -8\).
• Next, isolate the variable. Subtract 45 from each side, yielding \(6x \leq -53\).
• Divide both sides by 6, leading to \(x \leq -\frac{53}{6}\).

One key difference with inequalities is that if you multiply or divide by a negative number, you must flip the inequality sign. Always remember this rule to solve inequalities accurately.
Solving inequalities involves these steps of simplification and isolation, and once complete, you convert the result to a form that can be easily translated into interval notation.

Graphing Inequalities

Graphing inequalities on a number line gives a visual representation of the solution set, making it easier to understand the range of solutions.

• Begin by marking the significant point, which in this example is \( -\frac{53}{6} \).
• Since \( x \leq -\frac{53}{6} \), you use a closed dot at \( -\frac{53}{6} \) to show the number is included in the solution.
• Draw an arrow extending to the left from \( -\frac{53}{6} \) to represent all values less than or equal to this number.

This graph visually communicates which parts of the number line satisfy the inequality. It's a simple yet powerful tool for interpreting inequality solutions, clarifying which numbers are part of the solution set and how they are connected.