Question

Solve the inequality \(27-x \geq 5 x+3 .\) Write the solution in both set notation and interval notation.

Short answer

The solution is \( \{ x \mid x \leq 4 \} \) in set notation and \( (-\infty, 4] \) in interval notation.

Step-by-step solution

- Simplify the Inequality

Starting with the given inequality: \[ 27 - x \geq 5x + 3 \]First, let's get all the terms involving x on one side and the constant terms on the other side.

- Move x-terms to One Side

Add x to both sides of the inequality:\[ 27 - x + x \geq 5x + x + 3 \]Simplify to:\[ 27 \geq 6x + 3 \]

- Move Constants to One Side

Subtract 3 from both sides to isolate terms involving x:\[ 27 - 3 \geq 6x + 3 - 3 \]Simplify to:\[ 24 \geq 6x \]

- Solve for x

Divide both sides by 6 to solve for x:\[ \frac{24}{6} \geq \frac{6x}{6} \]Simplify to:\[ 4 \geq x \]or equivalently:\[ x \leq 4 \]

- Write the Solution in Set Notation

In set notation, the solution is:\[ \{ x \mid x \leq 4 \} \]

- Write the Solution in Interval Notation

In interval notation, the solution is:\[ (-\infty, 4] \]

Key concepts

inequality solutions

Inequalities are expressions that compare two values and show that one value is less than, greater than, less than or equal to, or greater than or equal to another value. Solving inequalities involves finding all possible values of the variable that make the inequality true. Here are some important steps to solve inequalities:

• Isolate the variable on one side of the inequality.
• Perform operations, such as addition, subtraction, multiplication, or division, on both sides to keep the inequality balanced.
• Remember, if you multiply or divide by a negative number, you must reverse the direction of the inequality sign.

The goal is to simplify the inequality to a form where the variable is clearly compared to a number. In this exercise, the inequality is simplified to: \( x \leq 4 \). This indicates that any value of x that is 4 or less satisfies the inequality.

set notation

Set notation is a way to describe a set of numbers that satisfy a certain condition. When we express solutions using set notation, we use a set builder form. For an inequality like \( x \leq 4 \), the set notation is written as:

\( \{ x \mid x \leq 4 \} \)

• The curly brackets \( \{ \} \) are used to denote a set.
• The vertical line \( \mid \) means 'such that'.
• The portion \( x \leq 4 \) represents the condition that must be satisfied.

This notation provides a clear and concise way to state that x can be any value as long as it is less than or equal to 4.

interval notation

Interval notation is another method to represent sets of numbers that satisfy an inequality. This notation uses intervals to show the range of possible values.

For the solution \( x \leq 4 \), the interval notation is:
\( (-\infty, 4] \)

• The parentheses - \( ( \) and \( ) \), - indicate that the endpoint is not included.
• Square brackets - \( [ \) and \( ] \) - indicate that the endpoint is included.
• The symbol \( -\infty \) means negative infinity, extending to the left indefinitely.

Thus, \( (-\infty, 4] \) denotes all values from negative infinity up to and including 4.

algebraic manipulation

Algebraic manipulation involves using various algebraic methods to simplify expressions or solve equations and inequalities. Key steps include:

• Combining like terms: Bringing similar terms together for simplification.
• Transposing terms: Moving terms from one side of an equation or inequality to the other, typically involving addition or subtraction.
• Isolating the variable: Getting the variable by itself on one side to simplify solving for it.
• Handling coefficients: Dividing or multiplying both sides by the coefficient of the variable to solve for it.

In our exercise, we followed these steps:

• Initially, we combined like terms and transposed terms involving x.
• Subtracted constants to isolate variable terms.
• Divided by the variable's coefficient to solve the inequality.

Algebraic manipulation is essential for solving inequalities like \( 27 - x \geq 5x + 3 \).