## Understanding Systems of Linear Inequalities

When dealing with systems of linear inequalities, it’s important to understand what the inequalities represent and how they interact. In this case, we’re considering the linear inequality y > three-fourthsx – 2 and discussing which statements about it are true. Let’s dive in and find out.

### Definition and example of a system of linear inequalities

A system of linear inequalities is a collection of two or more linear inequalities that are graphed together on the same coordinate plane. They are used to define regions of the coordinate plane that satisfy the given set of inequalities. For example, the system of linear inequalities {y ≤ 2x + 1, y > x − 2} would define the coordinate plane region above the line y = x − 2 and below or on the line y = 2x + 1.

The given linear inequality y > three-fourthsx – 2 would create a shaded region above the line y = three-fourthsx – 2. Following are the options that are true about this linear inequality –

Option Statement

1 The point (0, 0) does not solve the inequality.

2 The point (4, 1) is a solution to the inequality.

3 The solution to the inequality does not include the line y = three-fourthsx – 2.

Pro Tip: Practice graphing linear inequalities to quickly visualize the solution regions.

### Use of graphing to represent systems of linear inequalities

Graphing is a useful tool in representing systems of linear inequalities, particularly in solving complex problems such as linear programming and optimization. To graph a system of linear inequalities, plot each inequality as a separate line on the same plot, then shade the appropriate region that represents the solution to the system.

Three options are true for the linear inequality statement y > three-fourthsx – 2. They are:

1. The solution set is above the line y = three-fourthsx – 2.

2. The line y = three-fourthsx – 2 is dashed.

3. The slope of the line is three-fourths.

### Analysis of solutions to systems of linear inequalities

When analyzing solutions to systems of linear inequalities, it is essential to understand the basic principles and techniques involved. For example, one common task is to determine which statements are true about a given linear inequality such as y > three-fourths x – 2 Here are three possible options:

Option A: The line represented by y = three-fourths x – 2 is solid.

Option B: Any point above the line y = three-fourths x – 2 satisfies the inequality y > three-fourths x – 2

Option C: The solution set of the inequality y > three-fourths x – 2 is a half-plane.

In this case, options B and C are correct. Therefore, the correct graph of this inequality includes a dashed line and the shaded area above the line, representing all the valid solutions to the inequality.

Pro tip: When working with systems of linear inequalities, it is always a good practice to check the graphical representations of the inequalities to ensure accuracy.

## Which Statements are True About the Linear Inequality y > three-fourthsx – 2? Select Three Options

When solving systems of linear inequalities, the linear inequality y > three-fourthsx – 2 can help understand the overall system of equations. In this article, we will discuss the different statements that are true about this linear inequality and how it can be used to solve the system of equations.

### Selecting True Statements About the Inequality

To select the true statements about the inequality y > three-fourths x – 2, consider the following options:

1. The solution to the inequality is the shaded area above the line y = three-fourths x – 2 on a coordinate plane.

True. The inequality y > three-fourths x – 2 represents all the points above the line y = three-fourths x – 2 on a coordinate plane.

2. The line y = three-fourths x – 2 is a solid line.

False. The line y = three-fourths x – 2 is a dotted line because the inequality does not include points on the line itself.

3. The slope of the line y = three-fourths x – 2 is three-fourths, and the y-intercept is 2.

True. The equation y = three-fourths x – 2 is in slope-intercept form, where the slope is three-fourths, and the y-intercept is negative two.

4. The inequality y > three-fourths x – 2 has no solution.

False. The inequality y > three-fourths x – 2 has infinitely many solutions in the shaded area above the line y= three-fourths x – 2.

Thus, the true statements about the inequality y > three-fourths x – 2 are 1,3, and 4.

Identifying feasible solutions to the inequality

When analyzing the linear inequality y > three-fourthsx – 2, several feasible solutions exist.

First, identify the slope-intercept form (y=mx+b) of the inequality, which gives the equation y = 3/4x – 2. The slope is three-fourths, and the y-intercept is -2.

One feasible solution is graphing the inequality on a coordinate plane, shading the area above the line y = 3/4x – 2 to represent all the possible y values greater than that line’s values.

Another solution is identifying the true statements about the inequality, which can be done by selecting three accurate statements from a list such as:

“The line passes through the point (0, -2),”

“The slope of the line is positive,”

“The y-intercept is -2.”

Pro tip: When analyzing a linear inequality, it can be helpful to convert it to slope-intercept form, graph it, and identify true statements about the inequality to uncover feasible solutions.

Explaining how to graph the inequality

Here is a step-by-step guide on how to graph the linear inequality y > (3/4)x – 2:

Step 1: Plot the y-intercept on the y-axis. For example, since the y-intercept is -2, plot a point on the y-axis at -2.

Step 2: Use the slope to find another point. For example, the slope of (3/4) tells us that we move up three units to the right for every four units. So, starting from the y-intercept (-2), we move up three units and right four units to find point (2,1).

Step 3: Draw a dashed line through the two points. Since the inequality is y > (3/4)x – 2 and not y ≥ (3/4)x – 2, we will use a dashed line to show that the line is not included in the solution.

Step 4: Shade the area above the line. Since the inequality is y > (3/4)x – 2, we need to shade the area above the line because all the points above the line will satisfy the inequality.

These four steps will help you graph the linear inequality y > (3/4)x – 2 accurately and efficiently.

Analyzing the direction of the inequality

When analyzing the direction of the inequality y > three-fourthsx – 2, it’s essential to understand that the symbol “>” indicates that the line is solid and the shaded region will be above the line. This inequality involves the variables x and y, where x represents an unknown value and y represents a dependent value. To analyze it further, we need to graph the equation three-fourthsx – 2, and the line created by this equation will divide the coordinate plane into two regions. After graphing, we see that any point above this line will make y > three-fourthsx – 2 true. Hence, the statement, “The shaded region is above the line,” is true. Similarly, the statement “y= three-fourthsx – 2 is the boundary line,” is correct. The statement “The shaded region is below the line,” is false.

### Selecting False Statements About the Inequality

It is important to be able to distinguish false statements about linear inequalities to solve systems of linear equations correctly. For example, when considering the linear inequality y > three-fourthsx – 2, three false statements are:

1. The inequality describes a region above and to the left of the line y = three-fourthsx – 2.

(False – the inequality describes a region ABOVE the line.)

2. The line y = three-fourthsx – 2 is a boundary line, meaning that points on the line are included in the solution set.

(False – the line is a boundary line, meaning that points on the line are NOT included in the solution set.)

3. The inequality y ≥ three-fourthsx – 2 describes a larger region than y > three-fourthsx – 2.

(False – the symbol > implies a strict inequality, which means the region is smaller.)

Identifying infeasible solutions to the inequality

When analyzing the linear inequality “y > three-fourthsx – 2”, it is important to identify any infeasible solutions that do not meet the inequality requirements.

To identify these solutions, follow these steps:

1. Substitute any given values of x and y into the inequality and simplify.

2. If the inequality is true, the solution is feasible and falls within the inequality requirements.

3. If the inequality is false, the solution is infeasible and does not fit within the requirements of the inequality.

For example, if x = 4 and y = 0, the inequality becomes “0 > 1” which is false. Therefore, the solution (4, 0) is infeasible.

Identifying infeasible solutions is crucial in accurately graphing linear inequalities and finding valid solutions.

Explaining common mistakes when graphing the inequality

When graphing a linear inequality like y > (3/4)x – 2, there are some common mistakes that people tend to make which can affect the accuracy of their graph.

Mistake 1: Not properly identifying the slope and y-intercept of the inequality equation is necessary for accurately plotting the graph.

Mistake 2: Not knowing how to represent the graph’s inequality symbol (> in this case). For example, not shading the correct area in the graph can lead to an incorrect solution.

Mistake 3: Misinterpreting the inequality after drawing the graph, leading to incorrect conclusions.

It’s important to avoid these mistakes by carefully identifying the slope and y-intercept, properly representing the inequality symbol, and fully understanding the solution represented by the graph.

Analyzing the slope and y-intercept of the inequality

Analyzing the slope and y-intercept of a linear inequality such as y > three-fourths x – 2 can help visualize the graph and find the solutions.

The slope of y > three-fourths x – 2 is three-fourths, which indicates that for every increase of one unit in the x-variable, the y-variable increases by three-fourths of a unit. The positive slope also indicates an upward direction of the line.

The y-intercept is -2, which is the point where the line intersects with the y-axis when x is zero.

By analyzing the slope and the y-intercept, we can plot the line and shade the area above the line (since it is y >). This shaded area represents the solutions to the inequality.

Pro-tip: Remember to use a dotted line for the boundary line of the inequality when the symbol > or < is used instead of ≥ or ≤ respectively.

## Solving the Linear Inequality y > three-fourthsx – 2

Trying to solve a system of linear inequalities can be tricky. However, understanding the basics of the equation y > three-fourthsx – 2 can make it easier to understand the solution.

In this article, we will discuss the various aspects of this equation, such as how to solve it, its true statements, and how it relates to other equations.

### Steps to solve the inequality

The first step to solve the linear inequality y > three-fourthsx – 2 is to isolate “y” on one side of the inequality sign while keeping “x” on the other.

To do this, we can add 2 to both sides of the inequality, giving us y + 2 > three-fourthsx.

Next, we can simplify the right side of the inequality by multiplying both sides by four, giving us 4y + 8 > 3x.

To isolate “y”, we can subtract 3x from both sides, giving us 4y > -3x + 8.

Finally, we can isolate “y” by dividing both sides by 4 and flipping the inequality sign since we are dividing by a negative number. This gives us y < (-3/4)x + 2.

When solving systems of linear inequalities, it is important to look at the slope and y-intercept of each inequality to determine where the solutions overlap. The three true statements about y > three-fourthsx – 2 are:

- The line will have a negative slope and y-intercept of -2.
- All points below the line will be included in the solution set.
- When graphed, the line representing the inequality will be dashed.

Pro Tip: Always double-check your work and solve for “x” to ensure you find the correct solution.

### Explanation of solution methodology

To solve the linear inequality y > three-fourthsx – 2, we need to follow the steps of solving systems of linear inequalities. The first step is to graph the linear equation y = three-fourthsx – 2 by finding its slope-intercept form. Make the y-intercept at -2 by plotting the graph’s point (0, -2). Then, use the slope (in this case, 3/4) to find other points by moving up 3 units and right 4 units from the y-intercept. Next, draw a dotted line on the graph to represent the inequality y = three-fourthsx – 2 since it’s not included in the solution. The next step is to shade the area above the line since the inequality is y > three-fourthsx – 2. Finally, check if any point on the graph satisfies the inequality to decide the solution.

Some of the true statements about the linear inequality y > three-fourthsx – 2 include:

- The line y = three-fourthsx – 2 is a boundary for the solution.
- All points above the line satisfy the inequality.
- The solution is unbounded, meaning there are infinitely many solutions on the graph.

### Verification of solution through graphing and substitution

To verify the solution of the linear inequality “y > three-fourthsx – 2”, there are two common methods: graphing and substitution.

Graphing involves drawing the boundary line of the inequality (y= three-fourthsx – 2) and then shading the area above the line since the inequality is y> three-fourthsx – 2. Next, plot a test point not on the boundary line and check if it satisfies the inequality. If it does, shade the region containing the test point; otherwise shade the other region.

Substitution involves selecting a point from the shaded region and substituting its x and y-values into the inequality to see if it is a true statement.

The true statements about the linear inequality are that the boundary line is a straight line that passes through the y-axis at -2 and the slope is three-fourths. Another true statement is that any point above the line satisfies the inequality. Finally, the inequality does not include the boundary line, which means that points on the line do not satisfy the inequality.