Problem 98 - A Graphical Approach to College Algebra

Question

Solve each inequality analytically. Write the solution set in interval notation. Support your answer graphically. $$\frac{3}{4} x-0.2 x-6 \leq 0$$

Step-by-Step Solution

Answer

The solution set is \((-\infty, \frac{120}{11}]\).

Step 1: Combine Like Terms

First, we need to simplify the left side of the inequality \( \frac{3}{4}x - 0.2x - 6 \leq 0 \). Start by combining the \( x \)-terms. Convert \( 0.2 \) to a fraction to simplify: \( 0.2 = \frac{1}{5} \). Now combine \( \frac{3}{4}x - \frac{1}{5}x \). Find a common denominator, which is 20, resulting in \( \frac{15}{20}x - \frac{4}{20}x = \frac{11}{20}x \). Thus, the inequality becomes \( \frac{11}{20}x - 6 \leq 0 \).

Step 2: Isolate the Variable Term

Next, we want to isolate the term with \( x \) on one side. Add 6 to both sides of the inequality to get \( \frac{11}{20}x \leq 6 \).

Step 3: Solve for x

To solve for \( x \), multiply both sides by the reciprocal of \( \frac{11}{20} \), which is \( \frac{20}{11} \). This results in \( x \leq 6 \times \frac{20}{11} \). Multiplying, we get \( x \leq \frac{120}{11} \).

Step 4: Write the Solution in Interval Notation

The inequality \( x \leq \frac{120}{11} \) describes all values of \( x \) that are less than or equal to \( \frac{120}{11} \). In interval notation, this is written as \( (-\infty, \frac{120}{11}] \).

Step 5: Graph the Solution

To graph the solution, plot a number line. Mark \( \frac{120}{11} \), draw a closed circle at this point (because \( x \) is less than or equal to this value), and shade all numbers to the left of \( \frac{120}{11} \) towards \( -\infty \).

Key Concepts

Interval Notation

Interval notation is a way of expressing the set of solutions in a more compact form. Instead of listing all the potential solutions, it provides a concise way to represent them using intervals.
As per conventional practice:

• An open interval \(a, b\) represents all numbers greater than \(a\) and less than \(b\). Here, neither \(a\) nor \(b\) are included in the interval.
• A closed interval \[a, b\] includes both \(a\) and \(b\).
• Semi-open intervals, like \(a, b\] or \[a, b\), mean one endpoint is included while the other is not.

When we write \((-\infty, \frac{120}{11}]\), it signifies that the set includes all values from \(-\infty\) up to and including \(\frac{120}{11}\). The use of \(-\infty\) is a way to say the values extend left infinitely on the number line but don't include \(-\infty\) itself.

Combining Like Terms

Combining like terms is a fundamental simplification process used in algebra. It involves merging terms that have the same variable and power to simplify equations or inequalities.
In our exercise, we're dealing with terms like \(\frac{3}{4}x\) and \(0.2x\).

• First, convert decimals to fractions when needed. For example, \(0.2x\) converts to \(\frac{1}{5}x\).
• Once in fraction form, find a common denominator. Here, for \(\frac{3}{4}\) and \(\frac{1}{5}\), this is 20.
• After finding common denominators, adjust the fractions: \(\frac{3}{4}x\) becomes \(\frac{15}{20}x\) and \(\frac{1}{5}x\) becomes \(\frac{4}{20}x\).
• This lets us combine them: \(\frac{15}{20}x - \frac{4}{20}x = \frac{11}{20}x\).

This process simplifies expressions and makes it easier to further solve the inequality or equation.

Isolate the Variable

Isolating the variable is the technique used to get the variable on one side of the inequality or equation. This makes it possible to solve for the variable's value. Here's how it's done:

• Start by removing or canceling out constants and coefficients from the variable side. In our case, \(\frac{11}{20}x - 6 \leq 0\), we first add 6 to both sides to obtain \(\frac{11}{20}x \leq 6\).
• The next step is to remove the fraction. Multiply both sides by the reciprocal of \(\frac{11}{20}\). Here, you multiply by \(\frac{20}{11}\) to isolate the variable \(x\). This results in \(x \leq \frac{120}{11}\).

Through isolating the variable, the inequality transforms into a conclusion where \(x\) solely remains on one side. This clear expression is simpler to interpret and solve, presenting the specific range or value of \(x\).