Problem 84 - Algebra 1: Concepts and Skills

Question

Solve the inequality. Then graph and check the solution. $$ |x-3|>4 $$

Step-by-Step Solution

Answer

The solution to the inequality \(|x-3| > 4\) is \(x < -1\) or \(x > 7\).

Step 1: Solve for the Case Where (x-3) is Positive

The expression inside the absolute value brackets can be positive or negative. First, assume it's positive, and solve the inequality:\n\n\((x - 3) > 4\)\n\nSolving for \(x\), we get:\n\n\(x > 7\)

Step 2: Solve for the Case Where (x-3) is Negative

Second, assume the expression within the absolute value brackets is negative. To make (x - 3) negative, we take the opposite of (x - 3), which gives us - (x - 3). Now we solve this inequality:\n\n\(- (x - 3) > 4\).\n\nThis can be simplified to:\n\n\(- x + 3 > 4\).\n\nSolving for \(x\), we get:\n\n\(x < -1\).

Step 3: Graph and Check the Solution

The solutions \(x < -1\) and \(x > 7\) suggest that any \(x\) less than -1 or greater than 7 will satisfy the given inequality |x-3|>4. These solutions can be represented on a number line with open circles at x = -1 and x = 7, and shading to the left of -1 and to the right of 7 to indicate all numbers less than -1 and all numbers greater than 7. To check the solution, we can pick test points from each region. Verifying will confirm they satisfy the original inequality.

Key Concepts

Absolute Value

The absolute value of a number refers to its distance from zero on the number line, without considering direction. It is always a non-negative value. When we write the absolute value of a number, we use two vertical bars, like this: \( |x| \). For example, both \( |3| \) and \( |-3| \) are equal to 3 because each is three units away from zero.

In the context of algebra, absolute value can sometimes create two separate scenarios within an equation or an inequality, because the expression inside can be either positive or negative. This is why, for example, when we solve \( |x - 3| > 4 \), we consider both cases where \( (x - 3) \) is positive and where it is negative. Understanding this concept is crucial because it lays the foundation for solving absolute value inequalities.

Inequality Graphing

Graphing inequalities on a number line helps visualize the solution set. Unlike equations, where we often find a single number as a solution, inequalities have a range of solutions. When we graph an inequality, we use open circles to indicate that a number is not included in the solution set, and we use closed circles when a number is included.

For instance, if we were to graph the solutions to \( x > 7 \), we'd place an open circle at \( x = 7 \) and shade everything to the right, indicating all numbers greater than 7. On the other hand, if we have \( x \geq 7 \), we would use a closed circle, including 7 in our solution set. Including a graph in the solution helps verify that the range we provide encompasses all possible solutions and that we've correctly interpreted the inequality symbols.

Algebraic Solution

Finding an algebraic solution to an inequality means isolating the variable on one side of the inequality. To solve an inequality like \( |x - 3| > 4 \), we perform algebraic operations as we would in an equation, while remembering two main points:

• If we multiply or divide by a negative number, we must flip the inequality sign.
• We must consider the 'positive' and 'negative' scenarios separately for absolute values.

For the given inequality, we solve it in two steps, once assuming the inside of the absolute value (\( x - 3 \) in this case) is positive, and once assuming it is negative. By doing so, we are effectively removing the absolute value and turning it into two simple inequalities, thus finding the solution range for \( x \).

Number Line Representation

A number line is a visual representation that shows numbers as points on a line. In inequalities, the number line helps us depict the set of all possible solutions. For absolute value inequalities, the solution set can appear in two disjoint parts, depending on whether the numbers satisfy the inequality when the inside of the absolute value is positive or negative.

To represent the solution of \( |x - 3| > 4 \), we draw a number line, mark the points \( x = -1 \) and \( x = 7 \) with open circles (since these are not included in the solution), and shade the line to the left of \( x = -1 \) and to the right of \( x = 7 \). This clearly shows that all numbers less than -1 and greater than 7 will solve the inequality, and it conveys a lot of information at a glance—perfect for double-checking our algebraic solution.