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Chapter 10: Problem 56

Solve each inequality. Write the solution set in interval notation and graph it. $$ 3 x^{2}-243<0 $$

Short Answer

Expert verified
Solution set is \((-9, 9)\).

Step by step solution

01

Understand the Inequality

We need to solve the inequality \(3x^2 - 243 < 0\). First, we isolate the quadratic term by moving other terms to the other side of the inequality sign.
02

Isolate the Quadratic Term

Rewrite the inequality as \(3x^2 < 243\). Divide each side by 3 to isolate \(x^2\): \(x^2 < 81\).
03

Solve the Quadratic Inequality

The inequality \(x^2 < 81\) tells us that we are looking for values for \(x\) between two points. Solve \(x^2 = 81\) by taking the square root of both sides to find the boundaries: \(x = 9\) and \(x = -9\).
04

Determine Valid Solutions

The solution to the inequality \(x^2 < 81\) is when \(-9 < x < 9\). These are the values that make the original inequality true.
05

Express the Solution in Interval Notation

The solution set of the inequality \(3x^2 - 243 < 0\) in interval notation is \((-9, 9)\). This represents all real numbers between -9 and 9, not including -9 and 9.
06

Graph the Solution

Draw a number line. Mark the points -9 and 9 with open circles (since these are not included in the interval). Shade the region between -9 and 9 to indicate all numbers within this interval are solutions of the inequality.

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Key Concepts

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

Interval Notation
Interval notation is a concise way of expressing a range of values that are solutions to inequalities. It's like a shortcut that tells us which numbers lie between two endpoints.
For instance, in the solution \((-9, 9)\), the parentheses indicate open intervals, which means the endpoints -9 and 9 themselves are not included in the solution set. If the endpoints were included, square brackets \([-9, 9]\) would be used.
  • Parentheses \(( , )\): Excludes the endpoint.
  • Square Brackets \([ , ]\): Includes the endpoint.
This format simplifies the expression of all numbers that satisfy conditions within a specific range, making it easier to read and write solutions.
Graphing Inequalities
Graphing inequalities visually represents all possible solutions on a number line or coordinate plane. Let's use our inequality \(3x^2 - 243 < 0\) as an example.
When we graph this inequality on a number line, we only focus on the values that satisfy the condition. Here's how you can do it:
  • Identify key points from the solution \((-9, 9)\). In this case, they are -9 and 9.
  • Mark these points as open circles on the number line. Open circles mean the points themselves aren't solutions.
  • Shade the area between these circles. This represents every number in the interval \((-9, 9)\).
Graphing provides a clear visual representation, which helps us understand where the solutions lie on the number line.
Solving Quadratic Equations
Solving quadratic equations is essential for finding solutions to inequalities like \(3x^2 - 243 < 0\). A quadratic equation typically has the format \(ax^2 + bx + c = 0\). Here's a simple way to break it down:
  • In this exercise, the quadratic part is \(x^2\). Once isolated, we solve \(x^2 = 81\) by finding x-intercepts.
  • The roots of this equation are x = 9 and x = -9, these numbers solve \(x^2 = 81\).
  • Consider the inequality \(x^2 < 81\) which means x must land between -9 and 9.
By solving the quadratic equation, we determine the boundary points for our inequality, which helps us define the plot for graphing and express the interval notation precisely.

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Most popular questions from this chapter

Solve each equation. Approximate the solutions to the nearest hundredth when appropriate. $$ r^{2}-6 r-27=0 $$

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WRITING Describe the following solution set of a rational inequality in words: \((-\infty, 4] \cup(6,7)\)

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