Problem 57 - Algebra 1: Concepts and Skills

Question

Graph the inequality. \(y-4 x<0\)

Step-by-Step Solution

Answer

To graph the given inequality \(y-4x<0\), first rearrange the inequality to be in the slope-intercept form: \(y<4x\). Next, graph the line as if the inequality were an equals sign. The area below the line represents the solution to the inequality.

Step 1: Rearrange the inequality

Rearrange the inequality to get it in slope-intercept form, \(y=mx+b\). Here, we only need to transfer \(4x\) to the other side of the inequality. So we get: \(y<4x\)

Step 2: Graph the equation

Graph the line as if the inequality were an equals sign. Here, this is a straight line with a zero y-intercept (b=0) and a slope (m) of 4. So from origin (0,0), for every 'one step' movement to the right in x direction (x=1), y increases by 4 steps. Repeat this process to get multiple points to draw the line.

Step 3: Determine the shaded region

The inequality is less than 0, meaning we're looking for values of y that are smaller than 4x. This corresponds to the region below the line. Test this by selecting a point in the region (like the origin (0,0)) and confirm that it makes the inequality true.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to describe linear equations using the formula \(y = mx + b\). This form provides two key pieces of information about the line: the slope \(m\) and the y-intercept \(b\). The slope \(m\) tells you how steep the line is and the direction it goes. A positive slope means the line rises as it moves to the right, while a negative slope means it falls. The y-intercept \(b\) tells you where the line crosses the y-axis. Since the y-intercept is where \(x = 0\), it gives a starting point for drawing the line on the graph. Here are some general steps for graphing using the slope-intercept form:

• Start by plotting the y-intercept on the coordinate plane.
• Use the slope to find another point: from the y-intercept, move right one unit, then move up or down depending on the slope value.
• Draw the line through the points. For inequalities, determine if the line is solid (\(\leq\) or \(\geq\)) or dashed (\(<\) or \(>\)).

In the exercise, when \(y - 4x < 0\) is rearranged to \(y < 4x\), it tells us that our line is \(y = 4x\), with a slope of 4 and a y-intercept of 0.

Linear Inequalities

Linear inequalities are similar to linear equations, but they use inequality symbols like <, >, \(\leq\), or \(\geq\), instead of an equal sign. This introduces the concept of a range of solutions.

• A strict inequality like \(<\) or \(>\) indicates that the line itself is not included in the solution set.
• An inclusive inequality like \(\leq\) or \(\geq\) indicates that the line is part of the solution.

The solution to a linear inequality represents a region on the graph rather than just a line. This region is either above or below the line, depending on the inequality. For \(y < mx + b\), the region below the line is shaded, because the values of y are less than the line's value. To confirm the correct region is shaded, you can test a point in the shaded area to see if it satisfies the inequality, such as (0,0) in this case for \(y < 4x\).

Coordinate Plane

The coordinate plane is a tool used to visually represent algebraic equations and inequalities. It consists of two perpendicular lines called axes:

• The horizontal axis, known as the x-axis.
• The vertical axis, known as the y-axis.

The point where these axes intersect is known as the origin, labeled (0,0). Each point on the plane can be described by a pair of numbers (x, y), called coordinates. When working with inequalities, you plot points and lines on the coordinate plane to represent solutions visually.

Here’s a simple approach to graphing on a coordinate plane:

• Identify the y-intercept and plot it on the y-axis.
• Use the slope (rise over run) to find additional points on the line.
• Draw the line based on whether the inequality includes the line or not (dashed or solid).
• Choose a test point to confirm which side of the line represents the solution set and shade appropriately.

This systematic approach allows clear visualization of solutions and their respective regions for linear inequalities.