Problem 7 - College Algebra

Question

Solve the inequality. Write your final answer in interval notation. $$ 3 x+2 \geq 7 x-1 $$

Step-by-Step Solution

Answer

\((-\infty, \frac{3}{4}]\)

Step 1: Understand the Inequality

The given inequality is \( 3x + 2 \geq 7x - 1 \). Our goal is to find the values of \( x \) that satisfy this inequality.

Step 2: Move Terms Involving Variables to One Side

Subtract \( 7x \) from both sides of the inequality to get all terms involving \( x \) on one side:\[ 3x + 2 - 7x \geq -1 \]This simplifies to:\[ -4x + 2 \geq -1 \]

Step 3: Move Constant Terms to the Other Side

Subtract 2 from both sides to isolate the term involving \( x \):\[ -4x + 2 - 2 \geq -1 - 2 \]This simplifies the inequality to:\[ -4x \geq -3 \]

Step 4: Solve for \( x \)

Divide both sides of the inequality by \(-4\). Remember, dividing by a negative number reverses the inequality sign:\[ x \leq \frac{3}{4} \]

Step 5: Write the Solution in Interval Notation

The solution \( x \leq \frac{3}{4} \) in interval notation is:\[ (-\infty, \frac{3}{4}] \]

Key Concepts

Interval Notation

Interval notation is a concise way of expressing a range of values for which an inequality holds true. It is particularly useful for describing intervals on the number line where specific conditions apply to a variable. Here, the solution to the inequality is expressed as \[ (-\infty, \frac{3}{4}] \]

• Parentheses (): Used to indicate that the endpoint is not included in the interval.
• Brackets []: Used when the endpoint is included in the interval.

For example, if an inequality specifies that a variable is less than or equal to a value, as in \( x \leq \frac{3}{4} \), we use a bracket on the right, as seen here. Understanding interval notation is crucial as it provides a clear, standardized way of presenting the solution set for an inequality.Breaking it down, the interval \((-\infty, \frac{3}{4}]\) means that the variable \(x\) can be any number less than or equal to \(\frac{3}{4}\), up to and including \(\frac{3}{4}\), but no larger.
It covers all numbers from negative infinity to \(\frac{3}{4}\) on the number line.

Inequality Manipulation

Inequality manipulation involves a series of steps to simplify and solve inequalities. Much like solving equations, it requires strategically changing the inequality to find the values that satisfy it. The key operations in manipulation are:

• Adding or subtracting terms: This involves moving terms from one side of the inequality to the other, just as in equations. For example, if you transfer \(3x\) to \(7x\), it involves subtracting \(7x\) from both sides, which produces \(-4x + 2 \geq -1\).
• Dividing or multiplying by a number: Important to remember, when you divide or multiply both sides of an inequality by a negative number, the inequality sign must flip. For instance, dividing both sides by \(-4\) reversed \(\geq\) to \(\leq\).
• Simplifying the expression: Combine like terms to simplify the inequality, as any unnecessary complexity can lead to mistakes.

Successfully manipulating inequalities is essential for working mathematically with data, logic, or variable assessments. It requires care, especially with negative numbers, as missteps here can lead to reversing the intended results.

Variable Isolation

Variable isolation is about getting the variable by itself on one side of the inequality. This makes it much easier to determine the values of the variable that solve the inequality. Starting with \[3x + 2 \geq 7x - 1\],you want to isolate \(x\) by removing other terms from its side:

• Move all terms involving \(x\) to one side, typically the left. Subtraction or addition is used here to eliminate these terms from the opposite side.
• Bring constant terms to the other side. Again, use simple addition or subtraction to achieve this.

In our example, when you subtract \(7x\) from both sides first, then subtract \(2\), you leave the isolated \(-4x\) on one side. Completing this with division by \(-4\) to yield \(x \leq \frac{3}{4}\) allows us to write the solution in a clear and definitive form, ready for interpretation.Variable isolation is a fundamental step for finding exact bounds and ensuring that all solutions are correctly evaluated and accounted for.