Problem 6 - College Algebra

Question

Solve the inequality. Write your final answer in interval notation. $$ 4 x-7 \leq 9 $$

Step-by-Step Solution

Answer

The solution to the inequality is \((-\infty, 4]\).

Step 1: Isolate the Variable Term

To solve the inequality \(4x - 7 \leq 9\), begin by adding 7 to both sides in order to isolate the term with the variable. This gives us \(4x - 7 + 7 \leq 9 + 7\), which simplifies to \(4x \leq 16\).

Step 2: Solve for the Variable

Now, we need to solve for \(x\) by dividing both sides of the inequality by 4. This gives us \(\frac{4x}{4} \leq \frac{16}{4}\), which simplifies to \(x \leq 4\).

Step 3: Write the Solution in Interval Notation

The inequality \(x \leq 4\) means that \(x\) can be any number less than or equal to 4. In interval notation, this is expressed as \((-\infty, 4]\).

Key Concepts

Interval Notation

Interval notation is a way of expressing the set of solutions to an inequality using simple mathematical symbols. It helps to describe the range of values that a variable can take. In our example, the solution was expressed as \(( -\infty, 4 ]\). This notation tells us that \(x\) can take on any value less than or equal to 4.
Here is a quick breakdown of how to read interval notation:

• Parentheses \(()\) are used to denote that a boundary value is not included in the set. For example, \((a, b)\) means \(x\) is greater than \(a\) and less than \(b\).
• Brackets \([]\) indicate that a boundary value is included. \([a, b]\) means \(x\) is greater than or equal to \(a\) and less than or equal to \(b\).
• If the interval extends infinitely in either direction, the symbols \(-\infty\) and \(+\infty\) are used.

Understanding interval notation is crucial for communicating solutions concisely in mathematics.

Isolate Variable

Isolating the variable in an inequality means rearranging the inequality so that the variable appears by itself on one side. This is a common first step when solving inequalities or equations.
In the given problem, we had \(4x - 7 \leq 9\). To isolate \(x\), we needed to remove the \(-7\).

Steps to Isolate the Variable

• Add or subtract terms: Begin by adding or subtracting terms from both sides to get all variable terms on one side and constant terms on the other. For example, add 7 to both sides to cancel out \(-7\): \(4x - 7 + 7 \leq 9 + 7\).
• Resulting simplification: This gives \(4x \leq 16\), which isolates the term with the variable \(x\).

Isolation helps simplify the inequality, making it easier to perform further calculation steps.

Division in Inequalities

Division is often used to solve inequalities, especially after isolating the variable. In this process, we divide both sides of the inequality by the coefficient of the variable. It's important to remember a critical rule with inequalities: if you divide or multiply both sides by a negative number, you must reverse the inequality sign.
In the exercise, once we had \(4x \leq 16\), the next step was simple division:

• Divide both sides by 4, the coefficient of \(x\): \(\frac{4x}{4} \leq \frac{16}{4}\).
• This simplifies to \(x \leq 4\), since dividing by a positive number (4 in this case) does not change the inequality direction.

Pay close attention to the sign whenever you're dealing with negative coefficients during division or multiplication in inequalities.

Algebraic Expressions

An algebraic expression involves numbers, variables, and arithmetic operations showing the relationship between different quantities. Understanding how to manipulate these expressions is a core skill in solving inequalities.
The expression \(4x - 7\) consists of:

• Variables: A letter like \(x\) that represents a number whose value is not known yet.
• Constants: Numbers like 4 and 7 that have fixed values.
• Operators: Symbols such as \(-\) and multiplication (implied between 4 and \(x\)) that define the operation to perform.

Manipulating algebraic expressions often involves simplifying, factoring, and rearranging terms to make solving for a variable possible. This skill paves the way to efficiently tackle more complex mathematics problems.