Problem 29 - Elementary Algebra

Question

Graph the linear equations and inequalities. $$ 6 \leq x+4 \leq 7 $$

Step-by-Step Solution

Answer

Solution: The solution of the compound inequality is $2 \leq x \leq 3$. The graph of the inequality on the number line is shown below with closed circles at x = 2 and x = 3, and a line connecting the two circles, representing all values between 2 and 3, inclusive. ``` ___ _____(_)__(_)_____ 1 2 3 ```

Step 1: Write down the given inequality

The given inequality is: $$ 6 \leq x+4 \leq 7 $$

Step 2: Solve for x

To find the range of values for x that satisfy the inequality, we need to isolate x. We can do this by subtracting 4 from all three parts of the inequality. This will give us: $$ 6 - 4 \leq x + 4 - 4 \leq 7 - 4 $$ Which simplifies to: $$ 2 \leq x \leq 3 $$ So, the values of x that satisfy the inequality are between 2 and 3, inclusive.

Step 3: Graph the inequality on the number line

We will now graph the solution on the number line. Since the inequality includes the endpoints, we will use closed circles at x = 2 and x = 3. Then, draw a line connecting the two circles to indicate that all values between 2 and 3 are part of the solution as well. Here is the graph of the inequality: ``` ___ _____(_)__(_)_____ 1 2 3 ``` The closed circles at x = 2 and x = 3 indicate that these values are included in the solution, and the line between them represents all values of x between 2 and 3.

Key Concepts

Linear Equations

Linear equations form the backbone of algebra and are one of the first types of equations students learn to solve. They represent straight lines when graphed on a coordinate plane and take the general form of \(ax + b = c\). Here,

• \(a\) is the coefficient of \(x\)
• \(b\) is a constant
• \(c\) is the result or output.

In solving linear equations, the goal is always to isolate \(x\) to find its value. This typically involves adding, subtracting, multiplying, or dividing both sides of the equation by the same number. In our example, the linear equation component is seen when we isolate \(x\) after rearranging the equation from the inequality. This fundamental technique is crucial as it sets the stage for understanding more complex equations in algebra. Remember, each step should balance the equation just like a scale, maintaining equality.

Number Line Graphing

Number line graphing is a simple and effective way to visualize solutions to equations or inequalities. A number line is a straight line where each point represents a number. It's a tool that helps display a range of values or demonstrate understanding of number relationships.
For example, when graphing the inequality \(2 \leq x \leq 3\), we're showing all possible values that \(x\) can take. This includes whole numbers, fractions, and decimals within the specified range. In this particular problem, the endpoint values 2 and 3 are included, denoted by closed circles on the number line.
To draw a proper number line graph:

• Determine your scale and draw a horizontal line.
• Identify and mark your key numbers (like 2 and 3 in our example).
• Use closed or open circles to indicate whether endpoints are included or not.
• Draw a line connecting the relevant points to show all solutions within the range.

This visual method enhances comprehension and enables easier interpretation of algebraic solutions.

Mathematical Inequalities

Inequalities are mathematical expressions involving symbols such as \(>\), \(<\), \(\geq\), and \(\leq\). These symbols express the relationship between two quantities and are key in indicating that one quantity is larger, smaller, or equal with a range of numbers.
The inequality \(6 \leq x + 4 \leq 7\) suggests that the value of \(x + 4\) should be at least 6 but not more than 7. Solving it step by step involved using properties similar to those used in linear equations, ensuring all parts of the inequality are addressed equally. This systematic approach maintains balance and helps isolate \(x\).
When visualizing inequalities, it's crucial to recognize:

• The difference between strict (\(<\) and \(>\)) and non-strict inequalities (\(\leq\) and \(\geq\)).
• How endpoints are denoted on graphs, with strict inequalities using open circles and non-strict ones using closed circles.
• The range of values that satisfy the inequality, as exemplified by a filled segment on the number line between 2 and 3 for \(2 \leq x \leq 3\).

Understanding inequalities is essential as they form the basis for more advanced topics in mathematics like functions, limits, and optimization.