Problem 63 - Intermediate Algebra

Question

Solve each inequality. Graph the solution set and write it using interval notation. $$ \frac{x-7}{2}-\frac{x-1}{5} \geq-\frac{x}{4} $$

Step-by-Step Solution

Answer

The solution is \(x \geq 6\), written in interval notation as \([6, \infty)\).

Step 1: Eliminate the Fractions

To eliminate fractions, find the least common multiple (LCM) of the denominators: 2, 5, and 4. The LCM is 20. Multiply every term in the inequality by 20 to get rid of the fractions:\[ 20 \left(\frac{x-7}{2}\right) - 20 \left(\frac{x-1}{5}\right) \geq 20 \left(-\frac{x}{4}\right) \]This simplifies to:\[ 10(x-7) - 4(x-1) \geq -5x \]

Step 2: Distribute and Simplify

Distribute 10 and -4 into the parentheses:\[ 10x - 70 - 4x + 4 \geq -5x \]Combine like terms on the left side:\[ 6x - 66 \geq -5x \]

Step 3: Isolate the Variable

Add 5x to both sides to get all the x terms on one side of the inequality:\[ 6x + 5x - 66 \geq 0 \]Combine like terms:\[ 11x - 66 \geq 0 \]

Step 4: Solve for x

Add 66 to both sides to isolate the term with x:\[ 11x \geq 66 \]Then, divide every term by 11 to solve for x:\[ x \geq 6 \]

Step 5: Write the Solution in Interval Notation

The solution to the inequality \(x \geq 6\) in interval notation is:\[ [6, \infty) \]

Step 6: Graph the Solution

To graph the solution \(x \geq 6\), draw a number line and place a closed circle at 6 to indicate that 6 is included. Shade the line to the right of 6 to represent all values greater than or equal to 6.

Key Concepts

Interval Notation

Interval notation is a mathematical way of expressing the set of solutions to inequalities. It provides an efficient and clear way to describe the domain where a variable meets a specific condition. Let's break it down:

• When an interval includes the endpoint, we use square brackets [ ] to denote it. For example, \([6, \infty)\) indicates that the solution set includes 6.
• On the other hand, when the interval doesn't include the endpoint, parentheses ( ) are used. In the given exercise, infiniti is never a concrete number so it always has a parenthesis.
• Using infinity, like \((6, \infty)\), means the solutions keep increasing or decreasing without bound.

Breaking these down further with our exercise, we find that the solution to the inequality \(x \geq 6\) means x is greater than or equal to 6. Therefore, the interval is \([6, \infty)\). This notation clearly shows all numbers x that make the inequality true.

LCM (Least Common Multiple)

The least common multiple (LCM) is crucial for simplifying equations, especially those involving fractions. It allows us to eliminate the fractions by finding a common base. Here's how it works:

• First, identify all the denominators in the problem, like in the original exercise, we have: 2, 5, and 4.
• Then, find the smallest positive number that is a multiple of each denominator. This number is called the LCM.
• For 2, 5, and 4, the LCM is 20. This is because 20 is the lowest number that each of these numbers can "fit into," meaning they divide 20 evenly.

Multiplying through by the LCM simplifies each term, effectively "clearing" the fractions. This transforms the equation into a simpler form. In our example, after multiplying each term by 20, we get rid of the fractions, leaving an easier inequality to solve: \(10(x - 7) - 4(x - 1) \geq -5x\). LCM is a powerful tool in algebra that makes working with fractions more manageable.

Graphing Inequalities

Graphing inequalities provides a visual representation of the solution to an inequality. By depicting solutions on a number line, students can better understand which numbers fulfill the inequality's condition. Here's how to do it:

• Begin by identifying the boundary number, where the inequality becomes an equality. For this example, it happens at \(x = 6\).
• Next, place a point on the number line. If the boundary number is included in the solution (as in \(x \geq 6\)), use a closed circle. If the number is not included (like \(x > 6\)), use an open circle.
• Finally, shade the number line in the direction that satisfies the inequality. Since the inequality is \(\geq\), shade to the right from the boundary to express all numbers greater than or equal to 6.

Graphing inequalities helps students see solutions and understand how algebraic expressions relate to real numbers. It reinforces the connection between abstract concepts and their practical implications.