Problem 33 - Algebra 2

Question

Graph each inequality. $$ |x+y| > 1 $$

Step-by-Step Solution

Answer

The solution consists of the regions above the line \(y = -x + 1\) and below the line \(y = -x - 1\), not including the lines themselves.

Step 1: Understand the Absolute Value Inequality

The inequality \(|x + y| > 1\) indicates that the expression \(x + y\) must be greater than 1 or less than -1. This breaks down into two separate inequalities: \(x + y > 1\) and \(x + y < -1\). We will graph these two inequalities separately.

Step 2: Graph the Inequality \(x + y > 1\)

Start by rewriting \(x + y > 1\) as \(y > -x + 1\). This is the region above the line \(y = -x + 1\). Draw a dashed line for \(y = -x + 1\) because the inequality does not include the boundary (it's a 'greater than' not 'greater or equal to'). Shade the region above this line.

Step 3: Graph the Inequality \(x + y < -1\)

Rewriting \(x + y < -1\) gives \(y < -x - 1\). This inequality represents the region below the line \(y = -x - 1\). Again, draw a dashed line for \(y = -x - 1\) since the boundary is not included. Shade the region below this line.

Step 4: Combine Both Graphs

Combine the graphs of \(y > -x + 1\) and \(y < -x - 1\). The solution to the inequality \(|x+y| > 1\) is the region outside the area between the two dashed lines. This means we should shade the space above the line \(y = -x + 1\) and below the line \(y = -x - 1\), excluding any region in between the two lines.

Key Concepts

Absolute Value Inequalities

Absolute value inequalities like \(|x + y| > 1\) provide valuable insights into the behavior of expressions based on a number's distance from zero. The absolute value function, represented by vertical bars like \(|-|\), measures how far a number is from zero on the number line. When dealing with inequalities, absolute value inequalities describe regions where expressions fulfill certain conditions, either being greater or less than a specified value.

In the given exercise, the inequality \(|x + y| > 1\) translates to two separate inequalities:

• \(x + y > 1\)
• \(x + y < -1\)

Both represent different spatial regions on the coordinate plane. Instead of finding good and bad ideas, we are separating regions that conform or contradict the condition expressed by the inequality.

It is crucial to become adept at handling absolute value inequalities because they appear in a plethora of mathematical contexts, from simple algebra problems to more complex calculus equations. Understanding how these inequalities segment and map regions will significantly aid in tackling graphical problems.

Inequalities in Two Variables

Inequalities in two variables, such as \(x + y > 1\) and \(x + y < -1\), explore the relationship between two different variables, often representing them graphically. When graphing such expressions on a coordinate plane, each inequality divides the plane into distinct zones, either fulfilling or breaching the inequality's condition.

In general, an inequality like \(y > -x + 1\) describes all the points \((x, y)\) that lie above the line \(y = -x + 1\). Similarly, \(y < -x - 1\) corresponds to the points below the line \(y = -x - 1\). By understanding these regions:

• You're separating your graph into areas of interest where each inequality holds true or false.
• The boundaries of these areas (lines) are typically drawn as dashed, signaling that points on the line are not part of the solution set unless the inequality includes an equality (\(\ge\) or \(\le\)).

The ability to switch between symbolic and graphic representations of inequalities is an essential skill in mathematics, allowing for greater flexibility in problem-solving.

Graphing Linear Inequalities

Graphing linear inequalities requires a methodical approach to properly illustrate solutions on the coordinate plane. Unlike equalities, which result in straight lines, inequalities deal with regions above, below, or surrounding these lines, depending on the inequality sign.

In the solution for \(|x + y| > 1\), the graph involves:

• Rewriting the inequalities: starting with converting \(x + y > 1\) and \(x + y < -1\) into the usual form \(y > -x + 1\) and \(y < -x - 1\), to understand the slope and interception of lines.
• Drawing dashed lines for \(y = -x + 1\) and \(y = -x - 1\), highlighting that points on these lines are not included in the set.
• Shading the outside areas beyond these lines to represent the solution to \(|x + y| > 1\): the region above \(y = -x + 1\) and below \(y = -x - 1\).

The process of shading plays a crucial role, as it visually distinguishes the solution set. Always ensure that your shading approach accurately reflects the inequality's conditions, joining multiple inequality solutions if necessary.