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Chapter 8: Problem 58

Graph and write interval notation for each compound inequality. $$ 3>-x \geq-1 $$

Short Answer

Expert verified
The solution in interval notation is (-3,1] and the corresponding graph involves an open circle at -3 and a closed circle at 1, with shading in-between.

Step by step solution

01

Solve the inequality for x

The given compound inequality is $$3 > -x \ -x \ x \ -x \ x$$. To isolate x, the inequalities should be solved separately: 1. Simplify the right part: $$3 > -x$$ Multiply both sides by -1 (remember to reverse the inequality sign): \ $$-3 < x \ x > -3$$ 2. Simplify the left part: $$-x \ -1$$ Multiply both sides by -1 (reverse the inequality sign): \ $$x \ 1$$
02

Combine the inequalities

Combine the two simplified inequalities $$-3 < x \ 1$$ into a single compound inequality: $$-3 \ x \ 1$$. This is written as $$-3 < x \ x \ 1$$.
03

Write in interval notation

To write the solution in interval notation: Transform the inequality $$(-3 < x , x \ 1) $$ into interval notation: (-3,1] The number -3 is not included as we use a parenthesis, while 1 is included as an endpoint, shown by a square bracket.
04

Graph the interval

Graph the interval on a number line: 1. Draw a horizontal line and mark the points -3 and 1. 2. Place an open circle at -3 to show that it is not included. 3. Place a closed circle at 1 to show that it is included. 4. Shade the region between -3 and 1. The graph shows all x-values in the interval -3<x<=1

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Key Concepts

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

graphing inequalities
Graphing inequalities on a number line helps visually represent solutions to inequalities. When graphing an inequality like \(-3 < x \leq 1\)\:
  • First, draw a horizontal line representing all possible values of x.
  • Next, identify key points 鈥� in this case, -3 and 1 on the line.
  • Then, use an open circle for points not included in the inequality and a closed circle for included points. For \(-3 < x \leq 1\), place an open circle at -3 and a closed circle at 1.
  • Finally, shade the region between the points to show all possible solutions, indicating that any value between but not including -3 and including 1 is a solution.
Graphically representing inequalities makes it easier to see and understand the solution set.
interval notation
Interval notation is a concise way of writing subsets of real numbers. It uses brackets and parentheses to describe the range of values. For the inequality \(-3 < x \leq 1\)\:
  • The interval starts at -3 and ends at 1.
  • Since -3 is not included, we use a parenthesis: \(-3\).
  • Since 1 is included, we use a square bracket: \[1\].
This gives us the interval notation \((-3,1] \). This notation provides a quick way to describe the set of numbers satisfying the inequality.
solving inequalities
Solving inequalities involves finding the range of values that satisfy the inequality. Here鈥檚 a step-by-step approach:
  • Simplify each part of the compound inequality separately: For \((3 > -x \ and \ -x \geq -1)\), we solve each part independently.
  • For 3 > -x, multiply both sides by -1, which reverses the inequality: \(-3 < x\).
  • For -x \geq -1, multiply both sides by -1, also reversing the inequality: \(x \leq 1\).
  • Combine the results into one compound inequality: \(-3 < x \leq 1\).
Solving these inequalities requires careful handling of negative values and ensuring that inequality signs are correctly reversed when multiplying or dividing by negative numbers.

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Most popular questions from this chapter

Solve. $$ |3 x-5|=x $$

To prepare for Chapter \(9,\) review solving systems of equations using elimination (Section 4.3). Solve. [ 4.3] $$ \begin{aligned} &3 x-5 y=1\\\ &2 x+3 y=7 \end{aligned} $$

Find the domain of each function. $$ f(x)=\sqrt{9-x^{2}} $$

For \(f(x)\) as given, use interval notation to write the domain of \(f\) $$ f(x)=\sqrt{x-6} $$

Number of Diagonals. A polygon with \(n\) sides has \(D\) diagonals, where \(D\) is given by the function $$ D(n)=\frac{n(n-3)}{2} $$ Find the number of sides \(n\) if \(27 \leq D \leq 230\).
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