Question

Solve the inequality. Then sketch a graph of the solution on a number line. $$|x+2|-1 \leq 8 $$

Short answer

The solution to the inequality is \( x \in [-11,7] \)

Step-by-step solution

Understanding the absolute value

The absolute value |x+2| refers to the distance of x+2 from zero. It is always positive except zero. Any absolute value inequality can be expressed in terms of two inequalities, one representing the positive absolute value and the other the negative absolute value.

Setting up two inequalities

The given inequality |x+2|-1 <= 8 can be translated into two separate inequalities as x + 2 - 1 <= 8 and -x - 2 - 1 <= 8.

Solving the inequalities

Solving each inequality will result in x<=7 and x>=-11.

Interpreting the solution

Since both inequalities use less than or equal to (<=) signs, the value x can take any number between and inclusive of -11 and 7.

Sketch the solution on a number line

On a number line, illustrate this solution set by shading the interval from -11 to 7 including the end points.

Key concepts

Understanding Absolute Value

In mathematics, the concept of absolute value is crucial when dealing with distances and can often appear in various forms of equations and inequalities. Think of absolute value as a tool to measure how far a number is from zero on a number line. It's always a non-negative number because distance cannot be negative.

For example, the expression \( |x+2| \) indicates the distance of \( x+2 \) from zero. One of the unique characteristics of absolute value is that it transforms negative inputs into their positive counterparts. This means that \( |-3| = 3 \) and \( |3| = 3 \) both have the same absolute value even though they are on opposite sides of zero on the number line.

A very important aspect when solving absolute value inequalities is to recognize that they can be split into two separate scenarios – one where the expression inside the absolute value is positive (which is straightforward) and one where the expression inside is negative (which requires you to flip the inequality sign when removing the absolute value). Understanding this concept is key to tackling these types of mathematical problems.

Solving Inequality Solutions

When we are presented with an inequality such as \( |x+2|-1 \leq 8 \), our goal is to find all the values of \( x \) that make the inequality true. Inequalities can seem daunting, but they follow similar steps to solving equations, with the added layer of considering the direction of the inequality sign.

To manage absolute value inequalities, we separate them into two distinct inequalities. First, we consider the case where the inside of the absolute value is positive, which gives us one inequality. Then, we consider the case where it is negative, and we remember to flip the inequality sign. In the given problem, this results in \( x + 2 - 1 \leq 8 \) or \( x \leq 7 \) and \( -x - 2 - 1 \leq 8 \) or \( x \geq -11 \) after solving. The solution is the set of \( x \) values that satisfy both conditions. When the inequalities include 'less than or equal to' signs, \( \leq \), it signals that the extreme values are part of the solution set, which is important when interpreting and graphing the results.

Understanding the solution to an inequality involves recognizing that it represents a range or a section of numbers, rather than a single solution - a concept fundamental in grasping how inequalities differ from equations.

Number Line Graphing

The final step of solving an inequality often involves number line graphing, which is a visual representation of the range of solutions. This graphical method is instrumental for visual learners as it illustrates the abstract concept of inequalities into a clear picture.

In the context of our exercise, the number line graph would depict all the numbers between and including -11 and 7. To accurately represent this on a number line, a horizontal line is drawn with markers for -11 and 7. Then, we shade in the region between these two points, including the endpoints (since our inequality is inclusive). This shaded area represents all the possible values of \( x \) that make the original inequality true.

Drawing the Number Line

Start with a straight horizontal line. Place a dot or a 'closed circle' at both -11 and 7 to indicate that these numbers are included in the solution set. Then, shade the interval in between to highlight the range of valid solutions. This simple visual aid can immensely help in interpreting and understanding the solution to an inequality.