Problem 41 - College Algebra

Question

For the following exercises, graph both straight lines (left-hand side being \(y_{1}\) and right-hand side being \(y_{2}\) ) on the same axes. Find the poin of intersection and solve the inequality by observing where it is true comparing the \(y\) -values of the lines. $$ \frac{1}{2} x+1>\frac{1}{2} x-5 $$

Step-by-Step Solution

Answer

The inequality holds for all real numbers \(x\).

Step 1: Identify the equations

To solve this exercise, we need to understand that the left-hand side of the inequality represents one line, and the right-hand side represents another. The given inequality is:\[ \frac{1}{2}x + 1 > \frac{1}{2}x - 5 \]So, we have:\[ y_1 = \frac{1}{2}x + 1 \] \[ y_2 = \frac{1}{2}x - 5 \]

Step 2: Compute the intersection point

Since the lines have the same coefficient of \(x\), they are parallel and will not intersect. Therefore, there is no intersection point, and we can directly compare the constant terms to determine where the inequality holds.

Step 3: Compare y-values without graphing

Here we simply compare the constant parts of the two lines:- For \( y_1 \), the y-intercept is \(1\) (from \(\frac{1}{2}x + 1\)).- For \( y_2 \), the y-intercept is \(-5\) (from \(\frac{1}{2}x - 5\)).Since for every value of \(x\), \(1 > -5\), \(y_1\) is always greater than \(y_2\).

Step 4: Check where inequality holds true

Since \( y_1 > y_2 \) at every point due to their constant terms being different and the slope of lines being the same, the inequality \( \frac{1}{2}x + 1 > \frac{1}{2}x - 5 \) holds true for all real numbers \(x\).

Key Concepts

Intersection of Lines

When we talk about the intersection of lines, we are referring to the point where two lines meet or cross each other on a graph. In mathematical terms, this is the point where both lines have the same set of coordinates \(x,y\). If you have two different equations, such as \(y = m_1x + c_1\) and \(y = m_2x + c_2\), the lines intersect if there exists a value of \(x\) where both resulting \(y\) coordinates are the same.

• Slope: The slope of a line indicates how steep it is. If two lines have different slopes, they may intersect, assuming they extend infinitely.
• Y-Intercept: The y-intercept is where a line crosses the y-axis. If two lines have the same slope but different y-intercepts, they are parallel and never intersect.

To find the intersection of lines, you typically set the equations equal to each other and solve for \(x\). However, in our step-by-step solution case, the lines do not intersect because they are parallel.

Parallel Lines

Parallel lines are a fundamental concept in geometry and coordinate algebra. When two lines are parallel, they never meet no matter how far they extend. This property arises because they have the same slope (rate of change), which ensures they always remain the same distance apart.For example, given two lines:- Line 1: \(y_1 = \frac{1}{2}x + 1\)- Line 2: \(y_2 = \frac{1}{2}x - 5\)These lines have identical slopes of \(\frac{1}{2}\). Thus, they will never cross each other. Instead, they run parallel, meaning they do not have a point of intersection. When analyzing inequalities with parallel lines:

• If one line sits ‘above’ another (greater y-intercept), it will impact the inequality.
• Here, \(y_1 > y_2\) since \(1 > -5\) at any \(x\) value.

Linear Inequalities

Linear inequalities involve expressions that have a linear form but are not equations. Instead of solving for points where two lines meet, you find where one line is greater or less than the other.Consider the inequality from the exercise:- \(\frac{1}{2}x + 1 > \frac{1}{2}x - 5\)Because these lines are parallel, as discussed, it revolves around comparing their y-intercepts:

• Y-Intercept Values: \(y_1 = \frac{1}{2}x + 1\) means a y-intercept of 1.
• Y-Intercept Values: \(y_2 = \frac{1}{2}x - 5\) means a y-intercept of -5.
• Since 1 is always greater than -5, \(y_1\) is always greater than \(y_2\).

This means the inequality holds true for all values of \(x\). Solving linear inequalities often doesn't require exact solving but understanding the relationship between the lines.