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Chapter 2: Problem 102

Solve each equation and inequality. Write the solution set of each inequality in interval notation and graph it. $$ \text { a. }-16<4(x+8) \leq 8 \quad \text { b. }-16=4(x+8)+8 $$

Short Answer

Expert verified
a. Interval: \((-12, -6]\) Equation: \(x = -14\)

Step by step solution

01

Simplify each side of the inequality

For the inequality \[-16<4(x+8) \leq 8\]First, distribute the 4 to both terms inside the parentheses:\[-16 < 4x + 32 \leq 8\]
02

Solve the inequality

Subtract 32 from all parts of the inequality to isolate the term containing x:\[-16 - 32 < 4x \leq 8 - 32\]This simplifies to:\[-48 < 4x \leq -24\]Now, divide all parts by 4 to solve for x:\[-12 < x \leq -6\]
03

Write the solution set in interval notation

The solution set for the inequality is \(-12 < x \leq -6\),which in interval notation is:\((-12, -6]\)
04

Graph the solution set on a number line

Draw a number line. Open circle on -12 (not included), closed circle on -6 (included), and shade between them to represent the interval:\( ext{------(====鈿玗}\)where ... represents values less than -12, and === represents values between -12 and -6 inclusive.
05

Simplify the equation

For the equation \[-16=4(x+8)+8\]Distribute the 4 inside the parentheses:\[-16 = 4x + 32 + 8\]Combine like terms:\[-16 = 4x + 40\]
06

Isolate the variable

To solve for x, first subtract 40 from both sides of the equation:\[-16 - 40 = 4x\]This simplifies to:\[-56 = 4x\]Now, divide both sides by 4:\[-14 = x\]
07

Solution for the equation

The solution of the equation is \(x = -14\).

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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 mathematical way of expressing a range of values. It's often used to convey solutions to inequalities in a compact form. When using interval notation, you denote the beginning and end of an interval, indicating which values are included or excluded.
Here's how it works:
  • Round brackets, \( ( \) and \ ) \, are used for values that are not included in the interval (open interval).
  • Square brackets, \[ [ \] \ and \ ] \, are used for values that are included in the interval (closed interval).
  • The interval \((-12, -6]\) means -12 is not included in the solution set, while -6 is included.
Interval notation is a succinct way to describe a set of numbers. It eliminates the need for writing full descriptions and is perfect for concisely communicating the range of solutions for inequalities.
Number Line Graph
A number line graph is a visual representation of numbers on a straight line. This helps in clearly showing the solution set for an inequality. When graphing intervals:
  • An open circle represents a number not included in the interval (e.g., -12 in \((-12, -6]\)).
  • A closed circle represents a number that is included in the interval (e.g., -6 in \((-12, -6]\)).
  • A shaded region between the circles indicates all the numbers that are part of the solution set.
The graph offers a straightforward way to visualize the solution to an inequality. Because it shows both included and excluded values, it helps prevent misunderstandings about the solution set.
Distributive Property
The distributive property is a foundational principle used in algebra to multiply a single term across terms within parentheses. For example, when you see an expression like \(4(x+8)\), you distribute the 4 across both \(x\) and 8.
  • To apply the distributive property, multiply the term outside the parentheses by each term inside the parentheses: \(4(x+8) = 4x + 32\).
  • This property helps simplify expressions and equations, making them easier to solve.
  • By distributing before isolating variables, it ensures all components are accounted for in the simplification process.
Understanding the distributive property is crucial for working with all sorts of algebraic equations, as it allows for the simplification required to find solutions.

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