Problem 11 - Algebra and Trigonometry

Question

Solve the linear inequality. Express the solution using interval notation and graph the solution set. $$ 2 x \leq 7 $$

Step-by-Step Solution

Answer

The solution in interval notation is \((-\infty, \frac{7}{2}]\).

Step 1: Isolate the Variable

First, we need to isolate the variable \( x \) by dividing both sides of the inequality by 2. This yields the inequality \( x \leq \frac{7}{2} \).

Step 2: Convert to Interval Notation

Now that we have \( x \leq \frac{7}{2} \), we can express this in interval notation. The solution set is all values of \( x \) that are less than or equal to \( \frac{7}{2} \). In interval notation, this is written as \((-\infty, \frac{7}{2}]\).

Step 3: Graph the Solution Set

To graph the solution set, draw a number line. Indicate the point \( \frac{7}{2} \) and shade the region to the left of this point to denote \( x \leq \frac{7}{2} \). Place a closed dot on \( \frac{7}{2} \) to show that this point is included in the solution.

Key Concepts

Variable Isolation

When solving linear inequalities, one of the first steps involves isolating the variable. This is similar to solving linear equations where we want one variable to be by itself on one side of the inequality sign.
To isolate the variable, you may need to perform operations such as addition, subtraction, multiplication, or division on both sides of the inequality. The goal is to get the variable, in this case, \( x \), alone.

For the inequality given:

• Start with: \( 2x \leq 7 \)
• Divide both sides by 2 to solve for \( x \).
• This gives: \( x \leq \frac{7}{2} \).

Remember the rule about flipping the inequality sign only applies when multiplying or dividing by a negative number. Since 2 isn’t negative, the inequality direction remains the same.

Interval Notation

Once the variable is isolated, expressing the solution in interval notation can clearly communicate which values satisfy the inequality.
Interval notation is a way to describe sets of numbers along a number line. Here’s how it works:

• Use a bracket \([\) or \(]\) to indicate that an endpoint is included.
• Use a parenthesis \((\) or \()\) to indicate that an endpoint is not included.
• Use \( \infty \) or \( -\infty \) to indicate unbounded intervals.

For the solution \( x \leq \frac{7}{2} \), all the values less than or equal to \( \frac{7}{2} \) are part of the solution set. This is expressed in interval notation as

• \(( -\infty, \frac{7}{2}]\)

Here, \(-\infty\) means there is no lower bound, and \(\frac{7}{2}\) is included in the solution, so it gets a closed bracket.

Number Line Graphing

Graphing the solution on a number line provides a visual representation of all the possible solutions for the inequality.
To graph \( x \leq \frac{7}{2} \), you should:

• Draw a horizontal number line.
• Mark the point \(\frac{7}{2}\) on this line.
• Shade the region extending from \(-\infty\) up to and including \(\frac{7}{2}\). This shows all numbers less than or equal to \(\frac{7}{2}\).
• Place a closed dot at \(\frac{7}{2}\) to indicate that this value is part of the solution set.

This graphical representation helps confirm the set of numbers that fulfill the inequality, making it easier to comprehend and confirm your results at a glance.