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Chapter 8: Problem 14

Graph the solution set, and write it using interval notation. $$ 5 x+6<76 $$

Short Answer

Expert verified
The solution set is \( (-\infty, 14) \).

Step by step solution

01

Isolate the variable

Move the constant to the other side by subtracting 6 from both sides of the inequality: \[ 5x + 6 - 6 < 76 - 6 \] Simplifying, we get: \[ 5x < 70 \]
02

Solve for x

Divide both sides by 5 to isolate x: \[ \frac{5x}{5} < \frac{70}{5} \] Simplifying, we get: \[ x < 14 \]
03

Write the solution in interval notation

The solution of \( x < 14 \) can be written in interval notation as: \((-\infty, 14)\)
04

Graph the solution set

To graph the solution set, draw a number line. Mark 14 and draw an open circle around it to indicate that 14 is not included. Shade everything to the left of 14 to represent all numbers less than 14.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Inequalities
When we graph inequalities, we're visually representing a range of values that satisfy the inequality.
In this exercise, our goal is to graph the solution for the inequality \(5x + 6 < 76\).
Once we simplify it to \(x < 14\), we can easily graph it. Here's what you'll need to do:
  • Draw a number line: This line includes all possible real numbers.
  • Identify the key point, which is 14 in our case: Place a point at 14.
  • Use an open circle at 14: This indicates that 14 is not included in the solution set (since we are dealing with <, not 鈮�).
  • Shade the line to the left: This shows that all values less than 14 satisfy the inequality.
By following these steps, you can create a clear visual representation of the solution to \(x < 14\).
Interval Notation
Interval notation is a concise way of writing the set of solutions to an inequality.
For the inequality \(x < 14\), we describe the range of values using interval notation.
Here's how it works:
  • The solution set includes all numbers less than 14.
  • Since 14 is not part of the solution, we use a parenthesis \( \) instead of a bracket \[ \].
  • We represent negative infinity with \((-\backslash infty\), indicating no lower bound.
  • The final interval notation for \(x < 14\) is \((-\backslash infty, 14)\).
This notation quickly communicates the range of solutions without needing a lengthy explanation. Just remember:
  • Parentheses \( \) indicate values not included in the set.
  • Brackets \[ \] indicate values that are included.
Algebraic Inequalities
Algebraic inequalities involve expressions on both sides of an inequality symbol (<, 鈮�, >, or 鈮�).
Let's break down solving the given inequality \(5x + 6 < 76\):
  • Isolate the variable: First, we need to get \(x\) by itself. We do this by subtracting 6 from both sides: \(5x + 6 - 6 < 76 - 6\) simplifies to \(5x < 70\).
  • Solve for \(x\): Next, divide both sides by 5 to isolate \(x\): \(\frac{5x}{5} < \frac{70}{5}\) which simplifies to \(x < 14\).
We now have our solution in its simplest form: \(x < 14\). This tells us that any value for \(x\) less than 14 will satisfy our original inequality.
Understanding these steps is crucial for solving any algebraic inequality. Practice isolating the variable and simplifying the expression to become more comfortable with these types of problems.

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Most popular questions from this chapter

Graph the solution set of each system of linear inequalities. $$ \begin{array}{r} 2 x+3 y<6 \\ x-y<5 \end{array} $$

Solve each compound inequality. Graph the solution set, and write it using interval notation. \(x+1>3\) or \(x+4<2\)

Graph each linear inequality. $$ x+4 y \geq-3 $$

Describe the region that is the solution set of each system of inequalities.$$ \begin{array}{l} x \geq 0 \\ y \geq 0 \end{array} $$

Graph the solution set, and write it using interval notation. $$ -3<\frac{3}{4} x<6 $$
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