Problem 36 - Algebra A Combined Function

Question

Solve each inequality. Write each answer using solution set notation. $$ 6 x<5 $$

Step-by-Step Solution

Answer

The solution in set notation is \(\{ x | x < \frac{5}{6} \}\).

Step 1: Understand the Inequality

The inequality given is \(6x < 5\). This means that the product of 6 and \(x\) must be less than 5.

Step 2: Isolate the Variable

To solve for \(x\), we need to isolate the variable on one side of the inequality. We can do this by dividing both sides of the inequality by 6.

Step 3: Divide Both Sides

Divide both sides of the inequality by 6 to isolate \(x\):\[\frac{6x}{6} < \frac{5}{6}\]This simplifies to:\[x < \frac{5}{6}\]

Step 4: Write the Solution in Set Notation

Now that we have isolated \(x\), we must express the solution using set notation. The solution set can be written as:\[\{ x | x < \frac{5}{6} \}\]

Key Concepts

Inequality Notation

Inequality notation is a way to express the relationship between two values where one is not necessarily equal to the other. In the given exercise, the inequality is represented as \(6x < 5\). This uses the less than symbol, "\(<\)", which shows us that the value on the left, \(6x\), is less than the value on the right, 5.
Inequalities can also use other symbols:

• \(>\) means greater than.
• \(\leq\) means less than or equal to.
• \(\geq\) means greater than or equal to.
• \(eq\) means not equal to.

These notations help to quickly convey different types of relationships between numbers, allowing for the concise presentation of a range of possible solutions.

Solution Set

A solution set is a collection of all possible values that satisfy a given inequality. In this problem, once we solve the inequality \(6x < 5\), we find that \(x < \frac{5}{6}\). This means that the solution to this inequality includes any value of \(x\) that is less than \(\frac{5}{6}\).
Using set notation, we can represent this solution set as \(\{ x | x < \frac{5}{6} \}\). This notation tells us that \(x\) can be any number less than \(\frac{5}{6}\). It's like a rule that \(x\) must follow.
A solution set is crucial because it provides a complete picture of all values that meet the criteria set by the inequality, allowing us to understand better the range of possible solutions.

Isolate the Variable

Isolating the variable is a key step when solving inequalities or equations. It involves getting the variable on one side of the inequality and everything else on the other side. In our example, we had \(6x < 5\). To solve for \(x\), we needed to get \(x\) all by itself on one side of the inequality.
To do this, we divided both sides of the inequality by 6, the coefficient of \(x\), which simplifies it to \(x < \frac{5}{6}\). This process involves performing the same mathematical operation on both sides of the inequality, which maintains the inequality's balance.
By isolating the variable, we derive a straightforward expression that indicates the range of values \(x\) can take. This makes it much easier to interpret the final solution and understand what \(x\) represents in the context of the inequality.