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Chapter 3: Problem 72

Solve each inequality by using the test-point method. State the solution set in interval notation and graph it. $$-5-s^{2}<0$$

Short Answer

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

Step by step solution

01

Simplify the Inequality

First, isolate the quadratic term. Start with the given inequality \( -5 - s^2 < 0 \). Add 5 to both sides to get \( -s^2 < 5 \). Next, multiply both sides by -1 (remember to reverse the inequality sign when multiplying or dividing by a negative number): \( s^2 > -5 \).
02

Analyze the Inequality

Since \( s^2 \) represents a squared term and is always non-negative, the inequality \( s^2 > -5 \) is always true for all real numbers \( s \). Thus, every value of \( s \) satisfies this inequality.
03

State the Solution Set

Since the inequality is true for all real numbers, the solution set in interval notation is \( (-\infty, \infty) \).
04

Graph the Solution Set

On a number line, shade the entire line to represent all real numbers.

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Key Concepts

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

quadratic inequality
A quadratic inequality is an inequality that involves a quadratic expression. Quadratic expressions have the general form \( ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants, and \( x \) represents the variable. These inequalities are used to express ranges of values that satisfy the quadratic condition, rather than a specific solution like a quadratic equation. Examples include \( x^2 - 4x + 3 > 0 \) or \( -5 - s^2 < 0 \). Quadratic inequalities can be solved using various methods, including factoring, completing the square, or using the Quadratic Formula. For our exercise, the inequality \( -5 - s^2 < 0 \) was given, and we simplified it to find a solution.
test-point method
The test-point method is a technique to determine the solution set of an inequality. To apply this method:
  • First, simplify the inequality to a standard form.
  • Find the critical points, usually by setting the expression equal to zero.
  • Divide the number line into intervals based on these critical points.
  • Select a test point from each interval and substitute it into the original inequality to check if it satisfies the inequality.
In the exercise provided, the inequality is simplified to \( s^2 > -5 \). Since \( s^2 \) is always non-negative, all values of \( s \) satisfy the inequality. This results in the solution set \( (-fty, fty) \).
interval notation
Interval notation is a way to represent the solution set of inequalities. Intervals show the range of values that satisfy the condition. Types of intervals include:
  • Open intervals: Use parentheses ( ), indicating the endpoints are not included. Example: \( (a, b) \).
  • Closed intervals: Use brackets [ ], indicating the endpoints are included. Example: \( [a, b] \).
  • Half-open/half-closed intervals: Combine a parenthesis and a bracket. Example: \( [a, b) \).
In our exercise, the solution to the inequality \( -5 - s^2 < 0 \) was all real numbers. Therefore, the interval notation is \( (-fty, fty) \), indicating every real number is included in the solution.
graphing inequalities
Graphing inequalities involves representing the solution set on a number line for one-variable inequalities or on a coordinate plane for two-variable inequalities. For our one-variable inequality:
  • Identify the solution set: Use techniques like the test-point method.
  • Shade the solution area: On a number line, shade the portion that includes all solutions. Use open or closed dots to indicate whether endpoints are included.
  • Label correctly: Ensure clarity by labeling intervals and points correctly.
In our example, the solution to \( -5 - s^2 < 0 \) was all real numbers, \( (-fty, fty) \). We shade the entire number line to represent this, reflecting that every value of \( s \) satisfies the inequality.

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