Problem 42 - Algebra and Trigonometry

Question

\(23-48\) Solve the inequality. Express the answer using interval notation. $$ 7|x+2|+5>4 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, \infty)\).

Step 1: Isolate the Absolute Value Expression

The first step to solving this inequality is to isolate the absolute value term \(|x + 2|\). Start with the original inequality: \[7|x + 2| + 5 > 4\]Subtract 5 from both sides to isolate \(7|x + 2|\):\[7|x + 2| > -1\].

Step 2: Analyze the Inequality of Absolute Value

Since the absolute value function \(|x + 2|\) is always non-negative (i.e., \(|x + 2| \geq 0\)), the inequality \(7|x + 2| > -1\) is true for all values of \(x\). This is because a non-negative number multiplied by 7 is never less than -1.Therefore, the solution includes all values of \(x\).

Step 3: Interval Notation

Express the solution as an interval. Since all values of \(x\) satisfy the inequality, the solution in interval notation is \((-\infty, \infty)\).

Key Concepts

Absolute Value

Absolute value is a crucial concept in mathematics that measures how far a number is from zero on the number line, regardless of direction. It is always non-negative since it represents distance. For any real number \(a\), the absolute value is expressed as \(|a|\).

• If \(a\) is positive, \(|a| = a\)
• If \(a\) is negative, \(|a| = -a\) since it converts the negative into positive
• If \(a\) is zero, \(|a| = 0\)

In the inequality problem, \(|x + 2|\) represents how far the expression \(x + 2\) is from zero, without considering its sign. This characteristic makes absolute value particularly useful for solving various mathematical equations and inequalities.
Understanding how absolute value works allows us to analyze expressions and solve inequalities that include absolute value terms.

Interval Notation

Interval notation provides a clear and concise way to represent a range of numbers. It shows all the numbers between two endpoints. Nailing down interval notation helps you communicate solutions precisely. These are some rules to keep in mind:

• Use parentheses \(( )\) for numbers that are not included in the interval, and brackets \([ ]\) for numbers that are included.
• The symbol \(-\infty\) or \(+\infty\) is always paired with parentheses because infinity is not a specific number that can be reached or included.

In the given exercise, we express all numbers from negative infinity to positive infinity as \((-\infty, \infty)\), meaning every real number is included. This solution indicates that there are no restrictions on \(x\) and every number satisfies the initial inequality. Understanding interval notation helps to easily interpret the solution sets of inequalities.

Inequality Solving

Solving an inequality means finding all values of the variable that make the inequality true. Like with equations, we aim to isolate the variable, but we also need to be mindful of inequalities' unique properties.

• When isolating terms, such as we had with the \(7|x+2|+5>4\), we start by getting the absolute value term by itself on one side of the inequality.
• Operations like addition or subtraction are performed similarly to solving equations.

A crucial step is understanding special properties, like absolute values, that influence inequality. In our example, isolating yields \(7|x+2| > -1\). Since absolute values are always zero or positive, \(7|x+2|\) is always greater than \(-1\). Thus, the inequality is satisfied by all real numbers.
Solving inequalities might seem challenging initially but breaking down each step, respecting the properties of operations and terms involved, will always lead you to the correct solution.