Problem 54 - Algebra for College Students

Question

Solve each inequality and express the solution set using interval notation. \(3(x-1) \geq-(x+4)\)

Step-by-Step Solution

Answer

The solution in interval notation is \([-\frac{1}{4}, \infty)\).

Step 1: Expand the inequality

Start by expanding both sides of the inequality. Distribute the 3 on the left side: \(3(x-1) = 3x - 3\). On the right side, distribute the negative sign: \(-(x+4) = -x - 4\). This gives us the inequality \(3x - 3 \geq -x - 4\).

Step 2: Isolate variable terms on one side

Add \(x\) to both sides to get all the \(x\) terms on one side: \(3x + x - 3 \geq -4\), simplifying to \(4x - 3 \geq -4\).

Step 3: Simplify the constant terms

Add 3 to both sides to move the constant term: \(4x - 3 + 3 \geq -4 + 3\), which simplifies to \(4x \geq -1\).

Step 4: Solve for the variable

Divide both sides by 4 to solve for \(x\): \(x \geq -\frac{1}{4}\). The solution to the inequality is \(x \geq -\frac{1}{4}\).

Step 5: Express in interval notation

Convert the solution \(x \geq -\frac{1}{4}\) into interval notation. Since \(x\) can be any value greater than or equal to \(-\frac{1}{4}\), the interval notation is \([-\frac{1}{4}, \infty)\).

Key Concepts

Interval Notation

When you solve inequalities, expressing the solution in interval notation plays a key role.
Interval notation is a concise way to describe a range of values that your variable can take.
In our inequality, we've solved for all the values of \(x\) that meet the condition \(x \geq -\frac{1}{4}\).

• Brackets [] or (): Use square brackets \([\;]\) for intervals that include an endpoint, meaning the value itself is part of the solution. Use parentheses \(()\) to indicate the value is not included.
• Using Infinity (∞): Always use a parenthesis around \(\infty\) because infinity is not a number you can reach, so it's not included.

Given the condition \(x \geq -\frac{1}{4}\), we include \(-\frac{1}{4}\) in our solution and continue to infinity. Thus, the interval notation is \([ -\frac{1}{4}, \infty )\).
This compact form clearly communicates the solution set.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging and simplifying equations or inequalities to isolate variables or simplify expressions.
This process is essential when solving inequalities or equations, as it helps clarify the relationships within the mathematical expression.In our example inequality \(3(x-1) \geq -(x+4)\), we performed several key steps:

• Distribute: We distributed coefficients across parentheses, turning \(3(x-1)\) into \(3x - 3\) and \(-(x+4)\) into \(-x - 4\).
• Combine Like Terms: Adding \(x\) to each side to gather terms with the variable \(x\) together resulted in \(4x - 3 \geq -4\).
• Shift Constants: Adding \(3\) to both sides moved the constant term, simplifying to \(4x \geq -1\).
• Isolate the Variable: Finally, dividing both sides by \(4\) solved for \(x\).

Understanding every step of algebraic manipulation strengthens your problem-solving skills and leads to valid solutions.

Inequality Solutions

Solving inequalities is a fundamental skill in algebra that involves finding all possible values for a variable that satisfy an inequality statement.
This concept is closely linked with algebraic manipulation and interval notation.To solve \(3(x-1) \geq -(x+4)\), we followed these general steps typical in solving inequalities:

• Expanding: Expressions are expanded through distribution, revealing more manageable terms.
• Recollection of Terms: All variables are collected on one side of the inequality, similar to solving an equation.
• Balancing the Inequality: Operations in inequality, like adding or subtracting terms, should maintain the inequality’s balance.
• Dividing or Multiplying: When dividing or multiplying by a negative number, it's crucial to reverse the inequality sign. In this specific example, this did not apply as we divided by a positive number.

In solving \(x \geq -\frac{1}{4}\), we found all \(x\) values satisfying the inequality are included in the solution set.
This understanding helps precisely determine a range of valid solutions.