Question

Check whether \((0,0)\) is a solution. Then sketch the graph of the inequality. $$ x-y \leq-2 $$

Short answer

The point \((0,0)\) is not a solution to the inequality \(x - y \leq -2\). The graph of the inequality is the area below the line \(y = x + 2\).

Step-by-step solution

Verify if the point is a solution

Substitute the coordinates of the point \((0,0)\) into the inequality. It becomes \(0 - 0 \leq -2\), simplifying to \(0 \leq -2\). This condition is false, therefore, \((0,0)\) is not a solution to the inequality.

Sketch the graph of the inequality

To sketch the graph, first, create the graph for the line \(x - y = -2\). Simplify this to \(y = x + 2\). This line intercepts the y-axis at \(2\) and has a slope of \(1\). Once the line graph is drawn, since \((0,0)\) is not a solution and it lies above the line, the area below the line should be shaded. This is to indicate the area of all the points that will satisfy the inequality.

Key concepts

Solving Inequalities

Solving inequalities is a fundamental part of algebra, closely related to solving equations. An inequality shows the relationship between two expressions that are not equal, indicating that one is larger or smaller than the other. When working with inequalities, you'll encounter symbols such as < (less than), > (greater than), \(\leq\) (less than or equal to), and \(\geq\) (greater than or equal to).

To solve an inequality means to find all the values of the variable that make the inequality true. This is commonly done by performing the same operations on both sides of the inequality without altering the original inequality's meaning.

However, there's a unique characteristic to keep in mind: when you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. For instance, if you have \( -2x > 4 \), dividing both sides by \( -2 \) would flip the sign to \( x < -2 \).

When it comes to verifying if a point is a solution to an inequality, simply substitute the x and y values into the inequality and see if you end up with a true statement. If the statement is true, the point lies in the solution region of the inequality.

Coordinate Graphing

Plotting Points and Lines

Coordinate graphing is an intuitive way to visualize relationships between numbers and equations on a two-dimensional plane. This plane is divided into four quadrants by a horizontal line (x-axis) and a vertical line (y-axis), with each point being represented by a pair of numbers, known as coordinates (x, y).

To graph an inequality, you must first plot the associated linear equation. This involves identifying points that satisfy the equation and then connecting these points to form a straight line.

Determining Which Side to Shade

For inequalities, one side of the line represents the solutions to the inequality. To determine which side to shade, you can pick a test point that's not on the line (often the origin \( (0,0) \) if it’s not part of the line). Substitute the test point's coordinates into the inequality. If the resulting statement is true, the test point's side is shaded; otherwise, the opposite side is shaded. This shaded area represents all the possible solutions to the inequality.

Slope-Intercept Form

The slope-intercept form is an equation of a line represented by \( y = mx + b \), where \( m \) is the slope of the line, and \( b \) is the y-intercept. The slope indicates the steepness and direction of the line, while the y-intercept indicates the point where the line crosses the y-axis.

Understanding Slope

The slope is calculated as the 'rise over run', which means the change in y divided by the change in x between two distinct points on the line. A positive slope means the line rises as x increases, while a negative slope indicates it falls as x increases.

Interpreting the Y-Intercept

The y-intercept, denoted by \( b \), provides a starting point for the line on the coordinate plane. It's where the line will intersect the y-axis, corresponding to the value of y when \( x = 0 \).

To graph a line using the slope-intercept form, start by plotting the y-intercept on the y-axis. Then, use the slope to determine the next points. For instance, a slope of \( 1 \) means that for each step to the right on the x-axis, you also move one step up on the y-axis. This simple method allows for quick and accurate graphing of linear equations and inequalities.