Problem 9 - Algebra and Trigonometry

Question

\(9-32\) me solve the linear inequality. Express the solution using interval notation and graph the solution set. $$ 2 x \leq 7 $$

Step-by-Step Solution

Answer

Solution: \((-\infty, \frac{7}{2}]\) with a solid dot at \(\frac{7}{2}\) on the number line.

Step 1: Isolate the variable

To solve the inequality \( 2x \leq 7 \), we first need to isolate \( x \). Divide both sides of the inequality by 2 to eliminate the coefficient next to \( x \). This gives us: \[ x \leq \frac{7}{2} \] This simplified inequality tells us that \( x \) can be at most \( \frac{7}{2} \).

Step 2: Express the solution in interval notation

With \( x \leq \frac{7}{2} \), the solution in interval notation is all the values up to and including \( \frac{7}{2} \). The interval notation is: \( (-\infty, \frac{7}{2}] \).

Step 3: Graph the solution set

To graph the solution \( x \leq \frac{7}{2} \), draw a number line. Shade all the area to the left of \( \frac{7}{2} \) (including \( \frac{7}{2} \) itself). Place a solid dot on \( \frac{7}{2} \) to indicate that \( \frac{7}{2} \) is included in the solution set.

Key Concepts

Solving Inequalities

Solving inequalities is similar to solving equations. The goal is to isolate the variable on one side. This gives a range of values that make the inequality true. In the exercise, we have the inequality \( 2x \leq 7 \). **Steps to Solving the Inequality:**

• Start by isolating the variable, which means getting "\( x \)" alone on one side of the inequality.
• For \( 2x \leq 7 \), divide each side by 2 to remove the coefficient of 2 from \( x \).
• This leaves us with \( x \leq \frac{7}{2} \).
• This solution tells us that any number less than or equal to \( \frac{7}{2} \) satisfies the inequality.

Remember, dividing or multiplying both sides of an inequality by a negative number reverses the inequality symbol. In this case, since we only divided by a positive number (2), the direction of the inequality symbol remains the same.

Interval Notation

Interval notation provides a clear way to describe a set of numbers along a number line. For the inequality \( x \leq \frac{7}{2} \), the set of numbers includes all values that are less than or equal to \( \frac{7}{2} \). **Understanding Interval Notation:**

• An interval uses brackets and parentheses to show where a set of numbers starts and ends.
• The interval \( (-\infty, \frac{7}{2}] \) means all numbers from negative infinity up to and including \( \frac{7}{2} \).
• The parenthesis "(" with "-\infty" indicates that infinity is not an actual number and cannot be included. "-\infty" suggests going infinitely to the left on the number line.
• The bracket "]" next to \( \frac{7}{2} \) shows that \( \frac{7}{2} \) is part of the solution.

Interval notation is ideal for concise solutions, especially when writing solutions for inequalities.

Graphing Inequalities

Graphing inequalities on a number line gives a visual representation of solutions. It shows exactly which numbers satisfy an inequality. For the inequality \( x \leq \frac{7}{2} \):**Steps for Graphing:**

• Draw a horizontal line to represent the number line.
• Place a point or a solid dot at \( \frac{7}{2} \) on this number line. The solid dot indicates \( \frac{7}{2} \) is part of the solution.
• Shade everything to the left of \( \frac{7}{2} \) to show all numbers less than \( \frac{7}{2} \) are included.
• No arrow is needed at the end of the shading because the solution extends indefinitely towards smaller numbers.

Graphing helps visualize solutions quickly, making it easier to understand inequalities' scope.