Question

Check whether \((0,0)\) is a solution. Then sketch the graph of the inequality. $$ x-y \leq 5 $$

Short answer

The point (0,0) is a solution to the inequality. In the graph, the line that represents the equation \(x - y = 5\) is drawn and the area below the line, including the line, is shaded.

Step-by-step solution

Check if (0,0) Satisfies the Inequality

Substitute x=0 and y=0 into the given inequality \(x - y \leq 5\) and check if the inequality holds. If it holds, then (0,0) is a solution to the inequality. The calculation would be as follows: 0 - 0 = 0 which is less or equal to 5. Hence, (0,0) is a solution to the inequality.

Draw the boundary line

Sketch the line \(x - y = 5\) which is the boundary line of the inequality. To sketch this line, isolate y to get the equation in slope-intercept form, \(y = x - 5\). The y-intercept is -5 and the slope is 1, therefore the line passes through the point (-5,0) and has a gradient of 1.

Determine and Shade the Region

Since (0,0) is a solution to the inequality, the region of the graph that includes this point should be shaded. Since the inequality is \(x - y \leq 5\), the shading is inclusive of the line, meaning points on the line are part of the solution. So, the area below the line is shaded.

Key concepts

Algebraic Inequalities

Understanding algebraic inequalities is a fundamental skill in algebra and helps with solving real-world problems. An inequality expresses a relationship where two values are not necessarily equal, and one is less than, greater than, less than or equal to, or greater than or equal to another.

The basic forms of inequalities include \textless, \textgreater, \textless=, and \textgreater=. These symbols stand for 'less than', 'greater than', 'less than or equal to', and 'greater than or equal to', respectively. For example, the inequality in the exercise, \(x-y \leq 5\), indicates that the difference between \(x\) and \(y\) is less than or equal to 5.

When you are checking if a point like \((0,0)\) is a solution to an inequality, you substitute the x and y values into the inequality. If the inequality holds true, then the point is a solution. In this case, \(0 - 0 = 0\) is indeed less than or equal to 5, indicating that the origin is a solution to the given inequality.

Graphing Inequalities

To visualize an inequality, graphing is an effective method. Graphing linear inequalities starts with graphing the boundary line. This line represents all the points that are exact solutions to the equality part of the inequality. If the inequality has a \(\leq\) or \(\geq\) sign, the boundary line is solid, showing that points on the line are included in the set of solutions.

For the inequality \(x - y \leq 5\), the boundary line is \(x - y = 5\). Once plotted, we determine which side of the line contains the solutions to the inequality. A practical way to figure this out is by selecting a test point, like \((0,0)\), which is not on the boundary line, and if it satisfies the inequality, then the region containing that point is shaded. Since it has been established that \((0,0)\) is a solution, the area below and including the line \(y = x - 5\) is shaded.

Slope-Intercept Form

One of the most commonly used forms to express the equation of a line is the slope-intercept form, written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The slope represents how steep the line is, while the y-intercept indicates where the line crosses the y-axis.

In our example, the boundary line of the inequality can be rewritten in slope-intercept form as \(y = x - 5\). Here, the slope (\(m\)) is 1, and the y-intercept (\(b\)) is -5, which means the line increases one unit up for every one unit it moves to the right, and it crosses the y-axis at \((0,-5)\). Plotting this line on a graph provides a clear picture of the relationship between \(x\) and \(y\), and helps us to illustrate the set of solutions for the inequality.