Problem 135 - Prealgebra and Introductory Algebra

Question

Given that \(c\) is an integer, find all values of \(c\) for which \(|c|<7\)

Step-by-Step Solution

Answer

The integers that satisfy the inequality \(|c|<7\) are -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6.

Step 1: Understanding Absolute Value

The expression \(|c|<7\) involves an absolute value, which represents the distance of the number \(c\) from zero on the number line. The distance from zero should be less than 7 units. This means that \(c\) can be a positive number less than 7, zero, or a negative number greater than -7.

Step 2: Setting the Inequality

We can break the absolute value expression into two separate inequalities. These inequalities are \(c<7\) and \(-c<7\).

Step 3: Solving the Inequalities

We solve the inequalities from step 2 separately. For the first, we find \(c\) less than 7. For the second inequality, we multiply both sides by -1 to find \(c>-7\).

Step 4: Finding Common Solutions

The solution set of the inequalities from step 3 is the intersection of their solutions, which means that \(c\) must satisfy both inequalities simultaneously. Since \(c\) is an integer, we list all numbers that lie between -7 and 7 exclusive.

Step 5: Listing the Solutions

The integers that satisfy the inequality \(|c|<7\) are -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6.

Key Concepts

Absolute Value

Absolute value is a fundamental concept that you will frequently encounter in math, especially when working with inequalities in prealgebra. It measures the distance of a number from zero on the number line, regardless of its direction. This means that both

• right-side numbers (positive values) and
• left-side numbers (negative values)

will be considered equal when using absolute values, simply because they share the same distance from zero.

For example, the absolute value of 5 is 5, denoted as \(|5| = 5\), and the absolute value of -5 is also 5, denoted as \(|-5| = 5\). In inequalities involving absolute values, like \(|c| < 7\), this concept tells us the possible distance from zero that our integer, \(c\), can be. This sets the foundation to find which values of \(c\) fit this condition.

Integer Solutions

When solving problems with inequalities, particularly those with absolute values, identifying integer solutions is crucial. Integer solutions are the whole numbers that can satisfy the inequalities set forth within the given conditions. In our problem, where \(|c| < 7\), we're specifically looking for integer values of \(c\) that fall within the specified range.

First, we determine our range from the absolute value inequality:- The inequality \(c < 7\) implies that \(c\) must be less than 7,- The inequality \(c > -7\) implies that \(c\) must be greater than -7.
When we consider integer solutions between these limits, \(c\) can take any whole number value that is greater than -7 and less than 7: \(-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5,\) or \(6\). Thus, these numbers are our integer solutions.

Number Line

The number line is a visual representation that assists in understanding and solving inequalities. It shows numbers as points on a line spaced evenly, helping us visualize the location and distance between them. When it comes to inequalities involving absolute values, like \(|c| < 7\), the number line becomes quite handy.

To represent the inequality on a number line, we first identify the boundary distances from zero—in this case, +7 and -7. We then mark all integers between -7 and 7, but not including 7 or -7 themselves. This visual cue facilitates an easier grasp of which integer solutions are valid, providing a clearer understanding of the range specified by the inequality. Seeing the range of integers from \(-6\) to \(6\) visually reaffirms that all these points satisfy \(|c| < 7\).

Solving Inequalities

Solving inequalities involves finding all values of a variable that satisfy the given condition. In our case with absolute values, the inequality \(|c| < 7\) is resolved by splitting it into two separate inequalities:

• c < 7
• c > -7

These two inequalities together describe a range.When working with inequalities involving absolute values, it's essential to recognize how splitting can simplify the process. Solving these individual inequalities permits us to identify the overlap between them, which in this example, translates to values greater than -7 and less than 7. We then look at whole number solutions (integers only in this scenario). This simple breakdown turns the often tricky task of dealing with absolute values and inequalities into a manageable process, ensuring we find all valid integer solutions.