Problem 40 - College Algebra

Question

Use interval notation to express solution sets and graph each solution set on a number line. Solve linear inequality. \(5(3-x) \leq 3 x-1\)

Step-by-Step Solution

Answer

The solution set is \(x \geq 2\) expressed in interval notation as [2, +infinity). It has been graphed on a number line with a closed circle at 2, and an arrow pointing to the right, towards positive infinity.

Step 1: Simplify the Equation

First, distribute the 5 through the parentheses to simplify the inequality: \(15 - 5x \leq 3x - 1\).

Step 2: Isolate the Variable

Next, isolate x by adding 5x to both sides and adding 1 to both sides of the inequality. This gives: \(15 + 1 \leq 3x + 5x \Rightarrow 16 \leq 8x \). From here, divide 8 from both sides to solve for x: \( x \geq 2\).

Step 3: Express Solution in Interval Notation

The solution \(x \geq 2\) can be expressed in interval notation as [2, +infinity).

Step 4: Graph the Solution

On a number line, indicate the interval with a closed circle at 2 (because it is included in the solution set) and an arrow pointing to the right, towards positive infinity, to cover all possible values greater than or equal to 2.

Key Concepts

Interval Notation

Interval notation is a concise way of representing solution sets for inequalities. It uses brackets and parentheses to describe intervals on the real number line. For example:

• Square brackets [ ] indicate that the endpoint is included in the interval.
• Parentheses ( ) signify that the endpoint is not part of the interval.

Naturally, in the solution to the inequality \(x \geq 2\), the interval notation is \([2, +\infty)\). The square bracket means that 2 is included in the solution, while the parenthesis shows that the interval extends indefinitely towards positive infinity.

Number Line

A number line is a visual tool used to represent numbers in order. It helps us understand where our solutions fall in the set of real numbers. For graphical representation, a number line helps you see the range of possible values satisfying an inequality. To graph the solution \(x \geq 2\), place a closed circle at 2 on the number line. The closed circle signifies that 2 is a part of the solution. Then, draw a continuous arrow extending to the right, which shows that all numbers greater than 2 are included.

Inequality Solution

In solving linear inequalities, the goal is to find the range of values that satisfy the inequality statement. Here's a simple process:

• Simplify both sides of the inequality.
• Isolate the variable by performing algebraic operations.
• Remember to reverse the inequality sign if you multiply or divide by a negative number - this doesn't apply in this example but is crucial for others.

In the case of \(5(3-x) \leq 3x-1\), simplifying and solving gives us \(x \geq 2\). This means all values of \(x\) greater than or equal to 2 make the inequality true.

Graphing Inequalities

Graphing inequalities is crucial for visualizing the solution set on a number line. It shows all possible solutions at a glance. Here's how to graph:

• Determine if your variable includes an endpoint: use a closed circle if it does, like in \(x \geq 2\).
• If the variable does not include an endpoint, use an open circle.
• Draw an arrow to indicate the direction in which the values increase or decrease.

For \(x \geq 2\), you'd use a closed circle at 2 and extend an arrow to the right, showing that all values 2 and above satisfy the inequality.