Problem 46 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the inequality and sketch the solution on the real number line. Use a graphing utility to verify your solutions graphically. $$3|4-5 x|<9$$

Step-by-Step Solution

Answer

The solution to the inequality \(3|4-5x|<9\) is \(0.2 < x < 1.4\)

Step 1: Isolate the Absolute Value

First, isolate the absolute value on one side of the inequality by dividing both sides of the inequality by 3, which gives: \(|4-5x|<3\)

Step 2: Split the Inequality

Next, the absolute value inequality \(|4-5x|<3\) would be split into two inequalities: \(4-5x<3\) and \(4-5x>-3\)

Step 3: Solve both inequalities individually

For the first inequality \(4-5x<3\), subtract 4 from both sides to get \(-5x<-1\). Then divide by -5 and remember to switch the inequality sign (since dividing or multiplying by a negative number switches the inequality sign), which gives \(x>0.2\). For the second inequality \(4-5x>-3\), subtract 4 from both sides to get \(-5x>-7\). Then divide by -5 to get \(x<1.4\)

Step 4: Combine both solutions

The solution to the inequality \(|4-5x|<3\) is the intersection of \(x>0.2\) and \(x<1.4\). These two ranges overlap from \(x=0.2\) to \(x=1.4\). Therefore, \(0.2 < x < 1.4\)

Step 5: Sketch the solution on the real number line

On a number line, draw an open circle at 0.2 and 1.4, then shade the area in between. Keeping the circles open indicates that the actual numbers \(0.2\) and \(1.4\) are not part of the solution.

Step 6: Verify using a graphing utility

Plot the function \(y=3|4-5x|\) and \(y=9\) to see their intersection points. The values of \(x\) for which \(y<9\) would be the solution to the inequality. It should match with the solution obtained earlier. The \(x\) values between \(0.2\) and \(1.4\) should make \(3|4−5x|<9\)

Key Concepts

Number Line Representation

One of the most effective ways to visualize the solution of an absolute value inequality is by using a number line. By drawing number lines, we essentially map the range of possible solutions clearly and concisely. In this scenario, once we derived the solution that the values of \( x \) lie between 0.2 and 1.4 (i.e., \(0.2 < x < 1.4\)), we proceed by sketching this on a number line.

To accurately represent this range on the number line:

• Start by marking the points corresponding to 0.2 and 1.4 on the line.
• Use open circles at both these points to indicate that 0.2 and 1.4 are not included in the solution.
• Shade the region between these two points to visually demonstrate the solution range.

This graphical representation helps us easily see which values of \( x \) satisfy the inequality, providing a quick and intuitive understanding.

Graphical Verification

After determining the numerical solution to an inequality, it's immensely valuable to confirm this solution using a graphical method. This is where graphing utilities, like graphing calculators or graphing software, come to the rescue. They allow us to visually check inequalities by plotting them.

In our exercise, we plot the expression \( y = 3|4 - 5x| \) alongside the horizontal line \( y = 9 \). The aim is to identify the region where the graph of \( y = 3|4 - 5x| \) lies below the horizontal line, signifying the solution to the inequality \( 3|4-5x| < 9 \). This graphical confirmation should match our earlier derived solution of \( 0.2 < x < 1.4 \).

Utilizing graphing tools not only affirms our calculated solution but also provides a visual insight into how and why the inequality holds true for certain values of \( x \).

Inequality Solution

The solution of an inequality involves determining a set of values that satisfy the given inequality statement. Solving absolute value inequalities starts with isolating the absolute value expression and then solving the resulting inequalities.

For the inequality \( 3|4-5x| < 9 \), our primary task is to divide both sides by 3 to isolate the absolute value, leading to \( |4-5x| < 3 \). From here, we split it into two "standard" inequalities: \( 4 - 5x < 3 \) and \( 4 - 5x > -3 \). Solving each of these gives:

• For \( 4 - 5x < 3 \): solving leads to \( x > 0.2 \).
• For \( 4 - 5x > -3 \): solving yields \( x < 1.4 \).

Combining these, the solution is \( 0.2 < x < 1.4 \). This systematic approach ensures a comprehensive understanding of where and how the inequality holds true.

Real Numbers

Real numbers encompass all the numbers that can be found on the number line, including both rational and irrational numbers. In the context of solving inequalities, this set of numbers provides the framework within which solutions are found.

When we solve an inequality like \( 3|4-5x| < 9 \), we are interested in finding the range of \( x \) that are real numbers satisfying the inequality condition. Here, the inequality solution \( 0.2 < x < 1.4 \) represents all real numbers between 0.2 and 1.4, not including the endpoints. These solutions reside on the continuum of the real number line.

Understanding real numbers is crucial because they form the "space" or context for drawing number lines and asserting whether specific solutions make sense in a real-world or theoretical scenario. Real numbers ensure that our solutions are practically applicable and correctly represented.