Problem 78 - 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. $$-2|4-x| \geq-4$$

Step-by-Step Solution

Answer

The solution to the inequality \(-2|4-x| \geq-4\) is \([2,6]\).

Step 1: Simplify the Inequality

Start by removing the negative sign so the inequality becomes \(|4-x| \leq 2\). The absolute value is less than or equal to 2, which means that it ranges from -2 to 2.

Step 2: Form Two Equations

The inequality is equivalent to two inequalities: \(4 - x \leq 2\) and \(4 - x \geq -2\).

Step 3: Solve the first equation, \(4 - x \leq 2\)

First isolate x, which leads to: \[-x \leq 2 - 4\] \[-x \leq -2\] Multiply every term by -1 and flip the inequality sign: \(x \geq 2\)

Step 4: Solve the second equation, \(4 - x \geq -2

Isolate x to get: \[-x \geq -2 - 4\] \[-x \geq -6\] Multiply every term by -1 and flip the inequality sign: \(x \leq 6\)

Step 5: Obtain Solution in Interval Notation

Combine the solutions from Step 3 and Step 4. The overlap for these two solutions (where both conditions are met) is \(2 \leq x \leq 6\). In interval notation, this is written as \([2,6]\).

Key Concepts

Interval Notation

Interval notation is a concise and efficient way to express a range of numbers that are solutions to an inequality. This method uses brackets and parentheses to indicate which numbers are included or excluded in that range.
When writing interval notation:

• An open parenthesis, ( or ), indicates that the boundary number is not included in the set.
• A square bracket, [ or ], indicates that the boundary number is included in the set.

In our exercise, after solving the inequality, we found that the range of values for which the inequality holds is from 2 to 6, inclusive. Therefore, using interval notation, it is written as \[ [2, 6] \]. Here, both 2 and 6 are included in the solution set, which is why we use square brackets.

Inequality Solving

Solving inequalities involves finding the set of values that satisfy the inequality condition. In this exercise, we began by transforming the absolute value inequality into two separate inequalities without the absolute value.
The original inequality \( -2|4-x| \geq -4 \) was simplified to \( |4-x| \leq 2 \). This transformation is crucial because absolute values express distance, meaning the expression inside the absolute value (\(4 - x\)) must lie between -2 and 2.

• The first inequality derived was \(4 - x \leq 2\), which we solved to get \(x \geq 2\).
• The second inequality was \(4 - x \geq -2\), yielding \(x \leq 6\).

After solving these inequalities, we looked for the overlapping solution. This overlap, where both conditions are true, is what defines the solution in terms of x. Thus, the solution was found to be \(2 \leq x \leq 6\), combining what we obtained from each inequality.

Number Line Graphing

Graphing the solution to an inequality on a number line provides a clear visual representation of the solution set. By marking numbers on a line, it's easy to see which values satisfy the inequality.
For the exercise, we first found the solution interval \[ [2, 6] \]. To graph this on a number line:

• Draw a horizontal line representing the real numbers.
• Mark the points corresponding to the boundaries, here 2 and 6.
• Place solid dots at 2 and at 6, indicating these points are included in the solution (since we use square brackets in interval notation).
• Shade the region between these dots, showing that all numbers in this range are part of the solution.

Through number line graphing, students can easily verify their solution and see that every number between and including 2 and 6 satisfies the original inequality.