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Chapter 1: Problem 29

Solve the given linear inequality. Write the solution set using interval notation. Graph the solution set. $$ -7

Short Answer

Expert verified
The solution set is \((-5, 3)\).

Step by step solution

01

Isolate x

The given inequality is \[-7<x-2<1.\] The goal is to solve for \(x\). To do this, add \(2\) to each part of the inequality to isolate the term containing \(x\):\[-7+2 < x-2+2 < 1+2.\] Simplifying each part gives:\[-5 < x < 3.\] This isolates \(x\) between \(-5\) and \(3\).
02

Write the Solution in Interval Notation

Now that the inequality \(-5 < x < 3\) is isolated, express the solution set using interval notation. The solution set is the open interval \((-5, 3)\). This interval includes all numbers greater than \(-5\) and less than \(3\), but does not include \(-5\) or \(3\) themselves.
03

Graph the Solution Set

Draw a number line to represent the solution set. Mark points at \(-5\) and \(3\) with open circles, indicating that these points are not included in the solution set. Shade the region between \(-5\) and \(3\) to show all the numbers that satisfy the inequality.

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Key Concepts

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

Solution Set
A solution set in mathematics refers to the collection of values that satisfy a given inequality or equation. In the context of the inequality \(-7
The original inequality can be manipulated by performing operations such as addition, subtraction, multiplication, or division to isolate the variable in question. In this case, we added 2 to each part of the compound inequality, arriving at \(-5 < x < 3\). This tells us that the values that make the inequality true lie between -5 and 3.

For this particular inequality, the solution set is described continuously, meaning every number in the range \(-5, 3\) is a valid solution. However, the endpoints themselves are not part of the solution, as indicated by the use of the open inequality signs \(<\).
Interval Notation
In mathematics, interval notation is a way of writing subsets of the real number line in a more compact form. It is especially useful for expressing solution sets of inequalities.

For the inequality \(-5 < x < 3\), the solution set was determined to lie between \(-5\) and \(3\). Using interval notation, this is written as \((-5, 3)\). The parentheses here indicate that neither endpoint is included, thanks to the strict inequality symbols, which means \(-5\) and \(3\) are not part of the solution set.

When working with inequalities, recognize these distinctions in interval notation:
  • "\([a, b]\)" includes both endpoints \(a\) and \(b\).
  • "\((a, b)\)" excludes \(a\) and \(b\), which is the case here.
  • If one endpoint is included, you will use a combination of brackets and parentheses, like \([a, b)\) or \((a, b]\).
Graphing Inequalities
Graphing inequalities is a visual method that provides insight into the range of values a variable can assume, which satisfies the inequality condition.

To graph the solution set of \(-5 < x < 3\) on a number line, follow these steps:
  • Draw a horizontal line representing all the real numbers.
  • Mark the specific point at \(-5\) and the point at \(3\) on this number line.
  • Instead of using solid dots, employ open circles at these points to depict that \(-5\) and \(3\) are not included in the solution set.
Now, shade or draw a line between these two open circles to indicate all the numbers between \(-5\) and \(3\). This shaded region visually represents every point that satisfies the inequality, illustrating that the variable \(x\) is greater than \(-5\) and less than \(3\).

Graphing inequalities helps one quickly comprehend the valid solutions and convey the nature of the solution set easily.

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Most popular questions from this chapter

Use factorization to simplify the given expression in part (a). Then, if instructed, find the indicated limit in part \((b)\). (a) \(\frac{2 x+10}{x^{2}+7 x+10}\) (b) \(\lim _{x \rightarrow-5} \frac{2 x+10}{x^{2}+7 x+10}\)

Find an equation of the circle that satisfies the given conditions. endpoints of a diameter at (4,2) and (-3,5)

The given algebraic expression is an unsimplified answer to a calculus problem. Simplify the expression. $$ \left(3 x^{2}+4 x-1\right)(4)(2 x-3)^{3}(2)+(2 x-3)^{4}(6 x+4) $$

Sketch the set of points in the \(x y\) plane whose coordinates satisfy the given inequality. \(1 \leq x^{2}+y^{2} \leq 4\)

Use factorization to simplify the given expression in part (a). Then, if instructed, find the indicated limit in part \((b)\). (a) \(\frac{x^{4}-5 x^{3}+4 x-20}{x^{4}-5 x^{3}+x-5}\) (b) \(\lim _{x \rightarrow 5} \frac{x^{4}-5 x^{3}+4 x-20}{x^{4}-5 x^{3}+x-5}\)
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