Problem 27 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ 9+\frac{1}{3} x \geq 4-\frac{1}{2} x $$

Step-by-Step Solution

Answer

The solution is \([-6, \infty)\).

Step 1: Eliminate Fractions by Finding a Common Denominator

The inequalities have fractions with denominators 3 and 2. The least common multiple of these is 6. Multiply every term of the inequality by 6 to eliminate the fractions: \[ 6 \left( 9 + \frac{1}{3}x \right) \geq 6 \left( 4 - \frac{1}{2}x \right) \] This simplifies to \[ 54 + 2x \geq 24 - 3x \].

Step 2: Combine Like Terms

To simplify, first add \(3x\) to both sides to keep all the \(x\) terms on one side:\[ 54 + 2x + 3x \geq 24 - 3x + 3x \].This results in: \[ 54 + 5x \geq 24 \].

Step 3: Isolate the Variable Term

Subtract 54 from both sides to isolate terms involving \(x\): \[ 54 + 5x - 54 \geq 24 - 54 \].This simplifies to: \[ 5x \geq -30 \].

Step 4: Solve for x

Divide both sides of the inequality by 5 to solve for \(x\):\[ \frac{5x}{5} \geq \frac{-30}{5} \].This gives us: \[ x \geq -6 \].

Step 5: Express the Solution in Interval Notation

The inequality \(x \geq -6\) in interval notation is written as: \[ [-6, \infty) \].

Key Concepts

Understanding Interval Notation

In mathematics, interval notation is a way of representing a set of numbers between two endpoints. It's an efficient and straightforward method to express solutions of inequalities.

Intervals can be open, closed, or half-open. Here’s what each term means:

• Open Intervals: Denoted with parentheses, like \( (a, b) \) and do not include the endpoints \(a\) and \(b\).
• Closed Intervals: Use square brackets, such as \( [a, b] \) and include both endpoints.
• Half-Open Intervals: Combine a bracket and a parenthesis. For example, \( [a, b) \) includes \(a\) but not \(b\).

When expressing solutions, you simply write the smallest number first and the largest number second, using the appropriate brackets to indicate whether these endpoints are included. In our exercise, the interval notation \( [-6, \infty) \) indicates that \(x\) is greater than or equal to \(-6\), extending to positive infinity without an upper bound.

Building a Foundation with Common Denominators

When dealing with fractions in equations or inequalities, it's often necessary to eliminate them to simplify the solving process. This can be achieved by finding a common denominator, which is a common multiple of the denominators you're working with.

Let's break it down:

• Identify the denominators in the equation or inequality. In our example, these are 3 and 2.
• Determine the least common multiple (LCM) of these denominators. The LCM of 3 and 2 is 6. \( 6 = 2 \times 3 \).
• Multiply all terms in the inequality by this common denominator. This step eliminates the fractions and simplifies subsequent calculations.

By multiplying every term in \(9+\frac{1}{3} x\geq 4-\frac{1}{2} x \) by 6, we effectively remove fractions, obtaining an easier expression to manage: \( 54 + 2x \geq 24 - 3x \).

Solving Inequalities Effortlessly

Solving inequalities is a fundamental skill in algebra. It involves finding the value of a variable that makes the inequality true. Here are some key steps:

• Isolate the Variable: Start by ensuring all terms with the variable are on one side of the inequality. This usually involves addition or subtraction.
• Simplify: Combine like terms on both sides if necessary to clean up the expression.
• Balance the Inequality: When manipulating the equation, such as multiplying or dividing, ensure you do the same operation to both sides.

In our problem, the inequality originally given was: \(54 + 2x \geq 24 - 3x\). By performing steps such as adding \(3x\) to both sides and subtracting the same number from each side, we arrive at \(5x \geq -30\). Finally, dividing through by 5 gives \(x \geq -6\). Remember: multiplying or dividing by a negative number flips the inequality sign, though this did not occur in this particular exercise.

The Art of Algebraic Manipulation

Algebraic manipulation involves reshaping an equation or inequality to aid in solving or simplifying the problem. This includes techniques such as using the distributive property, combining like terms, and adjusting equations. The goal is to simplify the problem effectively.

Here’s how you can apply these techniques:

• Use the Distributive Property: This property allows you to multiply a single term across terms inside a parenthesis, as shown when distributing 6 in our example: \(6(9 + \frac{1}{3}x)\).
• Combine Like Terms: Once the fractions were removed, we combined terms with \(x\) to form \(5x\).
• Simplify Algebraic Expressions: Subtract terms from both sides to isolate variables and bring clarity to the inequality.

Algebraic manipulation is essential in moving from a complex equation to a simple form, making it easier to identify solutions. Keep practicing these techniques to build confidence and proficiency in solving equations and inequalities.