Problem 11 - Precalculus Mathematics for Calculus

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

Solution is \((-\infty, 3.5]\). Graph: line extends left from 3.5 with a closed circle at 3.5.

Step 1: Isolate the Variable

To solve the inequality \(2x \leq 7\), we need to isolate \(x\). We start by dividing both sides of the inequality by 2 to keep the equation balanced. This gives us:\[ \frac{2x}{2} \leq \frac{7}{2} \]which simplifies to:\[ x \leq \frac{7}{2} \]or \(x \leq 3.5\).

Step 2: Write the Solution in Interval Notation

The solution \(x \leq 3.5\) means that \(x\) can be any number less than or equal to 3.5. In interval notation, this is expressed as \((-\infty, 3.5]\). This indicates the set of numbers from negative infinity up to and including 3.5.

Step 3: Graph the Solution Set

On a number line, you represent the solution \((-\infty, 3.5]\) by drawing a line or shading all values to the left of 3.5. Place a closed circle or dot on 3.5 to show that it is included in the solution set. The line should extend indefinitely to the left, indicating all numbers less than or equal to 3.5 are part of the solution.

Key Concepts

Interval Notation

Interval notation is a way to describe a range of numbers along the number line. This method is often used when expressing solutions to inequalities. Instead of listing all possible numbers, interval notation succinctly summarizes the span of values that satisfy the condition.
When you write something like \((2, 5)\), it signifies all numbers greater than 2 but less than 5. Parentheses indicate that the endpoints (2 and 5, in this case) are not included in the set.
In contrast, brackets are used to denote inclusion of an endpoint. For instance, \[3, 7\] includes both 3 and 7 along with all numbers in between. When the solution involves negative infinity or positive infinity, always use parentheses because infinity itself is not a number and cannot be included.
For example, with the inequality solution \((-infty, 3.5]\), infinity has a parenthesis and 3.5 a bracket because 3.5 is part of the solution.

Solving Inequalities

Solving inequalities is similar to solving equations, but with one key difference: inequalities show relationships of greater than or less than, rather than equality. To solve linear inequalities, follow these steps:

• Isolate the variable: Begin by getting the variable on one side of the inequality sign. This often involves adding, subtracting, multiplying, or dividing both sides by the same number.
• Use inverse operations: Just like equations, perform inverse operations to move any constants or coefficients.
• Be cautious with multiplication or division by negative numbers: If you multiply or divide by a negative number, the inequality symbol flips direction.

In the exercise \((2x \leq 7)\), dividing by 2 isolates the variable without reversing the inequality:
\[x \leq \frac{7}{2} \text{ or } x \leq 3.5.\]This solution tells us that x can be less than or equal to 3.5, ready for interval notation or graphing solutions.

Graphing Inequalities

Graphing inequalities on a number line is a visual way to represent solutions. Once you have solved an inequality and expressed it in interval notation, graphing becomes straightforward.
To graph the solution \((-infty, 3.5]\):

• Draw a number line. Focus on 3.5 as the critical point.
• Since \[3.5\] is included in the solution, place a closed or filled-in circle at 3.5.
• Shade all the numbers to the left of 3.5, extending the line indefinitely to signify inclusion of values less than 3.5.

This graphical method illustrates the solution set neatly, letting us understand that any value to the left of 3.5, and 3.5 itself, satisfies the original inequality.