Problem 26 - Algebra for College Students

Question

Solve each inequality. Graph the solution set, and write it using interval notation. \(\frac{5 x-6}{8}<8\)

Step-by-Step Solution

Answer

x < 14; Interval notation: (-∞, 14).

Step 1 - Clear the Fraction

Multiply both sides of the inequality by 8 to eliminate the fraction: \[\frac{5x - 6}{8} < 8\]\[8 \times \frac{5x - 6}{8} < 8 \times 8\]This simplifies to:\[5x - 6 < 64\]

Step 2 - Isolate the Variable

Add 6 to both sides of the inequality to isolate the term containing x:\[5x - 6 + 6 < 64 + 6\]This simplifies to:\[5x < 70\]

Step 3 - Solve for x

Divide both sides by 5 to solve for x:\[\frac{5x}{5} < \frac{70}{5}\]This simplifies to:\[x < 14\]

Step 4 - Graph the Solution

Graph the solution set on a number line. Draw an open circle at 14 and shade the region to the left of 14 to represent all values less than 14.

Step 5 - Write in Interval Notation

Express the solution set in interval notation. The interval notation for all values less than 14 is:\[(-\infty, 14)\]

Key Concepts

fraction elimination

When solving inequalities that involve fractions, the first step is usually to eliminate the fraction by multiplying both sides of the inequality by the denominator of the fraction. In our example: \(\frac{5x - 6}{8} < 8\)We multiply both sides by 8 to get rid of the fraction because multiplying by the denominator cancels it out. Here's how it looks:\[8 \times \frac{5x - 6}{8} < 8 \times 8\]This simplifies to:\[5x - 6 < 64\]You see, by multiplying both sides by 8, we have successfully removed the fraction from the inequality, making it easier to solve.

isolating variable

The next important step is to isolate the variable, which in this case is 'x'. To isolate 'x', we need to get rid of any constants that are on the same side as the variable. Here's where we are so far:\[5x - 6 < 64\]First, we eliminate -6 by adding 6 to both sides:\[5x - 6 + 6 < 64 + 6\]This simplifies to:\[5x < 70\]Next, we divide both sides by 5 to solve for 'x':\[\frac{5x}{5} < \frac{70}{5}\]This simplifies to:\[x < 14\]We've now isolated 'x' and solved the inequality.

interval notation

Interval notation is a concise way to represent a range of values. After solving the inequality, we use interval notation to express the solution set. For the inequality \(x < 14\), all values less than 14 are included.In interval notation, this is written as:\[(-\infty, 14)\]Here's what each part means:

• -\infty: Indicates that the values extend indefinitely to the left.
• ,: Separates the lowest and highest values in the interval.
• 14: The upper limit of our range.
• ( ): Parentheses mean that 14 is not included in the set.

Therefore, \[(-\infty, 14)\] represents all numbers less than 14.

graphing inequalities

Graphing inequalities visually shows the solution set on a number line. Here's how to graph the inequality \(x < 14\):1. Draw a number line.2. Locate the value 14 on the number line.3. Draw an open circle at 14 to indicate that 14 is not part of the solution.4. Shade the region to the left of 14 to represent all values less than 14.The open circle means 14 is not included, but any number to its left is part of the solution set. This visual representation is very helpful for understanding the range of possible solutions.