Problem 17 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the inequality and sketch the solution on the real number line. Use a graphing utility to verify your solution graphically. $$6 x>42$$

Step-by-Step Solution

Answer

The solution to the inequality \(6x > 42\) is \(x > 7\). On the number line, this is shown by an open dot at 7 with an arrow pointing to the right, indicating all numbers greater than 7.

Step 1: Solve the Inequality

Solving the inequality involves isolating the variable 'x'. Here the inequality is \(6x > 42\). To isolate 'x', the equation needs to be divided by 6 on both sides. This gives: \(x > 42/6\) which simplifies to \(x > 7\).

Step 2: Draw the Number Line

A number line is a line, on which every point shows a real number. Draw a straight line. Most number lines used in schoolwork are one-dimensional, running left to right.

Step 3: Indicate the Solution on The Number Line

On the number line, the solution \(x > 7\) means 'x' is greater than 7. So, put an open dot at 7 and draw an arrow to the right, indicating that 'x' is any number greater than 7.

Key Concepts

Real Number Line

Understanding the real number line is crucial for solving inequalities. Imagine the real number line as an infinite straight path where each point corresponds to a real number. The line is typically horizontal, stretching out indefinitely in both directions.

To visualize it:

• The left side represents smaller values, or negative numbers.
• The right side stands for larger values, or positive numbers.
• Every point on the line has an exact position corresponding to a real number.

The real number line is essential for graphically representing solutions to inequalities. For instance, when we solve the inequality \( 6x > 42 \), it translates to \( x > 7 \). We represent this solution on the number line from the point just greater than 7, continuing to the right, showing all numbers greater than 7.

Graphing Utility

A graphing utility is like a digital assistant for mathematics. It helps visualize mathematical concepts, including inequalities. By using software or online graphing tools, you can quickly and accurately plot equations and observe their behavior.

If you're dealing with \( 6x > 42 \), setting this up in a graphing utility can aid:

• Confirming the solution visually by plotting \( x > 7 \).
• Seeing the inequality displayed as a region or line on a graph.
• Understanding relationships between numbers in the context of the inequality.

Accessing these utilities is easy and they offer features beyond manual plotting, such as zooming in for better precision, helping to reinforce your understanding of variable relationships.

Isolation of Variables

Isolation of variables is a technique used in mathematics to solve equations and inequalities. It involves rearranging the equation so that one variable stands alone, making it easier to determine its value or range of values.

Here's how you isolate a variable in the inequality \( 6x > 42 \):

• Divide each side of the inequality by 6, which is the coefficient of \( x \).
• This gives \( x > 7 \) after simplification.
• Isolation helps us see that \( x \) must be greater than 7 to satisfy the inequality.

This process is straightforward but powerful because it removes extra factors, focusing solely on the variable of interest. Mastering this technique is key to solving many algebraic problems efficiently.