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Chapter 17: Problem 41

Graph the given inequalities on the number line. $$t \leq-5 \text { and } t \geq-5$$

Short Answer

Expert verified
The solution is \( t = -5 \); graph with a closed dot at \(-5\).

Step by step solution

01

Understand the Inequalities

The inequalities given are \( t \leq -5 \) and \( t \geq -5 \). This means that \( t \) is both less than or equal to \(-5\) and greater than or equal to \(-5\).
02

Determine the Solution Set

Since \( t \) must satisfy both conditions, it means \( t \) can only be \(-5\). This is because \(-5\) is the only value that is both less than or equal to and greater than or equal to \(-5\).
03

Graph the Solution on the Number Line

On the number line, represent \(-5\) with a closed dot or circle. The closed dot at \(-5\) indicates that \(-5\) is included in the solution set.

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Key Concepts

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

Number Line
A number line is a straight, horizontal line used in mathematics to represent real numbers. Each point on this line corresponds to a real number, arranged in increasing order from left to right.
This visual tool helps in identifying the position of a number and the relationship between numbers, such as inequalities or ranges.
When representing numbers, particular attention is given to how they are marked to indicate if a specific number is part of the solution.
For example:
  • Open dots are used to indicate that a number is not included (used for strict inequalities like \( t < -5 \)).

  • Closed dots show that a number is included (for non-strict inequalities like \( t \leq -5 \)).
When graphing the inequality \( t \leq -5 \) and \( t \geq -5 \), a closed dot is placed on the number line at \(-5\), illustrating that \(-5\) is a solution.
Solution Set
The solution set of an inequality is the collection of numbers that fulfill the condition expressed by the inequality.
For inequalities like \( t \leq -5 \) and \( t \geq -5 \), you first determine where both conditions apply simultaneously.
  • This means finding the values of \( t \) that make both inequalities true.

  • In this case, the equality signs direct you to the border point \(-5\).
Because \(-5\) is both less than or equal to, and greater than or equal to itself, it is the only number that satisfies both conditions.
Thus, the solution set is a single element, \(-5\).
In problems involving solution sets, it's critical to determine where conditions overlap.
When multiple inequalities are involved, always check if there is overlap or if singular values satisfy every condition.
Mathematical Inequalities
Mathematical inequalities are expressions involving the symbols \( <, \leq, >, \text{and} \geq \) to demonstrate the relative size or order of two values.
They are used to show how one quantity is bigger, smaller, or at least as big or small as another.
For instance:
  • \( x < 3 \) denotes that \( x \) can be any number smaller than 3.

  • \( x \geq 3 \) implies \( x \) is 3 or greater.

  • Compound inequalities, like \( a \leq b \leq c \), indicate \( b \) equals or lies between \( a \) and \( c \).
In the provided exercise, the inequalities \( t \leq -5 \) and \( t \geq -5 \) show that the only possible value for \( t \) which satisfies both conditions simultaneously is \(-5\).
Understanding and manipulating inequalities is essential in solving problems with constraints and identifying possible solutions.

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Most popular questions from this chapter

Solve the given problems. Draw a graph of the solution of the system \(y \geq 2 x^{2}-6\) and \(y=x-3.\)

Use inequalities to solve the given problems. Find an inequality of the form \(a x^{2}+b x+c < 0\) with \(a > 0\) for which the solution is \(-1 < x < 4\)

Draw a sketch of the graph of the given inequality. $$y \leq \sqrt{2 x+5}$$

Use a calculator to display the solution of the given inequality or system of inequalities. $$\begin{aligned} &y<3-x\\\ &y>3 x-x^{3} \end{aligned}$$

Draw a sketch of the graph of the given inequality. $$y \leq x^{3}-8$$
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