Problem 36 - College Algebra with Corequisite Support

Question

For the following exercises, graph the function. Observe the points of intersection and shade the \(x\) -axis representing the solution set to the inequality. Show your graph and write your final answer in interval notation. $$ |x-2|<7 $$

Step-by-Step Solution

Answer

The solution set is \((-5, 9)\).

Step 1: Rewrite the Inequality

The inequality can be rewritten without the absolute value by expressing it as two separate inequalities: \[ \begin{align*} x-2 &< 7, \ x-2 &> -7. \end{align*} \]

Step 2: Solve Each Inequality

Solve each of the inequality expressions. \[ \begin{align*} x - 2 &< 7 &\rightarrow x < 9 \ x - 2 &> -7 &\rightarrow x > -5 \end{align*} \] These solutions give us \(-5 < x < 9\).

Step 3: Graph the Inequality on a Number Line

On a number line, plot the values \(-5\) and \(9\), making open circles at these points because the inequality is strict (< symbol). Shade the region between \(-5\) and \(9\) to represent the solution set where the inequality \(|x-2| < 7\) is satisfied.

Step 4: Shade Region Below x-axis

Now, we model the solution on a coordinate plane. Draw a horizontal line below the \(x\)-axis from \(-5\) to \(9\) to indicate the solution set for \(x\), while maintaining the equality \(|x-2| < 7\). This line constitutes the shaded area without including the points \(-5\) and \(9\) themselves.

Step 5: Write Final Answer in Interval Notation

The solution in interval notation provides the range of \(x\)-values that satisfy the inequality. So, the final answer is \((-5, 9)\).

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve expressions where the absolute value symbol is used to denote the distance from zero on the number line. The expression \(|x-2| < 7\) means that the distance of \(x-2\) from zero is less than 7. To solve such inequalities, we reframe the problem into two separate linear inequalities without absolute value.

These separate inequalities correspond to the range of possible values for \((x-2)\) that maintain a distance less than 7 from zero. Thus, we write:

• \(x-2 < 7\)
• \(x-2 > -7\)

Solving these inequalities will reveal the range of numbers that make \(|x-2| < 7\) true. This results in the double inequality \(-5 < x < 9\). This double inequality indicates the set of all \(x\)-values that keep the original absolute value inequality satisfied.

Graphing

Graphing absolute value inequalities involves visualizing the solution set on a coordinate system. Once we've determined our inequality solution \(-5 < x < 9\), we can better understand the range of values by representing them graphically.

To do this on the coordinate plane, you can draw a horizontal line below the \(x\)-axis connecting the points \(-5\) and \(9\). This shaded line between \(-5\) and \(9\) visually represents the solution to the inequality \(|x-2| < 7\). It's crucial to note that since the inequality uses a less than sign \(<\), we don't include the endpoints, so these points are depicted as open circles on the graph.

Interval Notation

Interval notation provides a succinct way to represent the solution set of an inequality. Given our solution \(-5 < x < 9\), we translate this into interval notation as \((-5, 9)\).

The use of parentheses in \((-5, 9)\) indicates that the interval does not include the boundary points \(-5\) and \(9\). If the inequality were \(\leq\) or \(\geq\), indicating inclusive boundaries, we'd use square brackets \([\text{ or }]\) instead of parentheses.

Using interval notation efficiently communicates the range of solutions, especially when complemented by a graph. It simplifies the expression of all the values of \(x\) that satisfy a particular condition.

Number Line Representation

The number line provides a simple, visual means to represent and understand the solution set of an inequality. For the inequality \(|x-2| < 7\), we already determined that \(-5 < x < 9\). To display this on a number line:

• Plot the numbers \(-5\) and \(9\) as open circles. Open circles are used because the points themselves are not included in the solution set.
• Shade the segment between these two points to show the range of \(x\) that satisfies the inequality.

This representation helps to clearly visualize which numbers x can take, giving a direct way to interpret the outcome of solving the inequality. It's a helpful tool for confirming that our solution makes sense relative to the problem's constraints.