Problem 83 - College Algebra

Question

Solve each inequality by first rewriting each one as an equivalent inequality without absolute value bars. Graph the solution set on a number line. Express the solution set using interval notation. $$4+\left|3-\frac{x}{3}\right| \geq 9$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x \in [-\infty, -6] \cup [24, \infty]\)

Step 1: Writing the inequality without absolute value bars

To write the inequality without the absolute value bars, redefine the inequality in its equivalent range form. For an equation \(|a| \geq b\), it can be redefined as \(a \geq b\) or \(a \leq -b\). So, \(4 +|3 - x/3| \geq 9\) becomes \(3 - x/3 \geq 9 - 4\) or \(3 - x/3 \leq -(9 - 4)\).

Step 2: Simplifying the inequalities

Next, simplify the inequalities from step 1 to isolate the x. The first inequality simplifies to \(3 - x/3 \geq 5\), hence \(x/3 \leq 3 - 5\) resulting to \(x \leq 3(3 - 5)\). Solve for \(x\) to get \(x \leq -6\). The second inequality simplifies to \(3 - x/3 \leq -5\), hence \(x/3 \geq 3 + 5\) resulting to \(x \geq 3(3 + 5)\). Solve for \(x\) to get \(x \geq 24\).

Step 3: Expressing the solution set

Express the solution set as an interval on the real number line. Since \(x\) can be less than or equal to \(-6\) and greater than or equal to \(24\), the values of \(x\) that make the original inequality true are \(x \in [-\infty, -6] \cup [24, \infty]\).

Step 4: Graphing the solution set

To graph the solution set, plot on the number line the two intervals \([-6, -\infty]\) and \([24, \infty]\). Mark the points at -6 and 24 with a filled dot, and draw arrows pointing leftwards from -6 and rightwards from 24 indicating towards negative and positive infinity respectively.

Key Concepts

Absolute Value Inequalities

Absolute value inequalities deal with values that reflect distance from zero on a number line. This means any absolute value inequality will have a solution in the form of a range. Consider the inequality \(|a| \geq b\). It translates to two possible situations: either \(a \geq b\) or \(a \leq -b\). These conditions are key, as they represent both sides of the number line.

In the example given, the equation \(4 + |3 - \frac{x}{3}| \geq 9\) is first broken into two separate inequalities by removing the absolute value bars. This results in two scenarios: \(3 - \frac{x}{3} \geq 5\) and \(3 - \frac{x}{3} \leq -5\). These expressions link the idea of distance away from a central point, expressed in terms of \(x\). Understanding these two branches is crucial when working with absolute value inequalities.

Interval Notation

Interval notation is a way to describe the set of solutions for inequalities. It uses brackets and parentheses to show ranges of values. Closed brackets [ ] include the endpoint in the solution, while open parentheses ( ) exclude it.

For example, when the solution for \(x \leq -6\) and \(x \geq 24\), we express this in interval notation as \([-fty, -6] \cup [24, \infty]\).

In interval notation:

• The square bracket [ or ] indicates that the endpoint is included in the set.
• The round parenthesis ( or ) shows the endpoint is not part of the solution set.

Learning this notation makes it much easier to succinctly express complex sets of numbers.

Graphing Inequalities

Graphing inequalities involves representing the solutions on a number line. This provides a visual method to understand which values satisfy the inequality.

For the exercise we discussed, you would draw a number line and:

• Place a filled dot on \(-6\) and shade the line to the left, representing \(x \leq -6\).
• Place another filled dot on \(24\) and shade the line to the right, representing \(x \geq 24\).
• This graph will also feature arrows pointing towards negative infinity and positive infinity, ensuring the complete set is shown.

By doing this, graphing becomes not only a method of solving but also verifying your solutions, allowing for a clear view of all possibilities that satisfy the inequality.