Problem 16 - Algebra for College Students

Question

Solve each equation and inequality. \(|x+2|=9\)

Step-by-Step Solution

Answer

The solutions are x = 7 and x = -11.

Step 1: Understand Absolute Value

The expression \(|x+2| = 9\) requires us to understand the concept of absolute value. The absolute value of a number is its distance from zero on the number line, disregarding the sign. Thus, \||x| = 9\ means that \(|x| can be either 9 or -9\).

Step 2: Rewrite as Two Equations

The equation \(x + 2 = 9\) or \(x + 2 = -9\) is derived from \(|x + 2| = 9\). This is because a distance of 9 units from zero can occur in both positive and negative directions.

Step 3: Solve the First Equation

First, consider \(x + 2 = 9\). Subtract 2 from both sides to solve for \(x\).\[\begin{align*}x + 2 &= 9 x &= 9 - 2 x &= 7\end{align*}\]So, one solution is \(x = 7\).

Step 4: Solve the Second Equation

Next, consider \(x + 2 = -9\). Again, subtract 2 from both sides to find \(x\).\[\begin{align*}x + 2 &= -9 x &= -9 - 2 x &= -11\end{align*}\]Thus, another solution is \(x = -11\).

Step 5: Combine Solutions

Since both equations \((x + 2 = 9)\) and \((x + 2 = -9)\) provide valid solutions for the absolute value equation, we combine them. The solutions are \(x = 7\) and \(x = -11\).

Key Concepts

Solving Equations

Solving an equation involves finding the value of variables that make the equation true. In the case of an absolute value equation like \(|x+2|=9\), we focus on understanding the absolute value and then forming two separate equations. Absolute value represents the distance from zero, thus leading to two possible scenarios: one positive and one negative. For our example, the two equations are \(x+2=9\) and \(x+2=-9\). Solving each equation separately helps us find the values that satisfy the original absolute value equation.

• With \(x+2=9\), solve by subtracting 2 from both sides, giving \(x=7\).
• For \(x+2=-9\), subtract 2 from both sides to yield \(x=-11\).

Both solutions, \(x=7\) and \(x=-11\), are valid as they satisfy the original equation. Always remember, solving absolute value equations requires considering both the positive and negative solutions because they represent a distance from zero.

Algebraic Expressions

An algebraic expression involves variables, constants, and arithmetic operations. In dealing with absolute value, it's crucial to rewrite the expression so that it can be solved easily.To solve \(|x+2|=9\), we convert it into two simple algebraic expressions: \(x+2=9\) and \(x+2=-9\). These expressions involve understanding how absolute values are calculated and manipulating the variable \(x\) by isolating it on one side of the equation.

• For instance, \(x+2=9\) involves simply subtracting 2 from each side, letting us isolate \(x\).
• Similarly, treat \(x+2=-9\) in the same manner by subtracting 2 to obtain the correct value for \(x\).

Understanding these manipulations and transformations not only simplifies the problem but also equips you with the skills to handle a variety of complex algebraic challenges.

Inequalities

Inequality is a fundamental concept in algebra, though it is slightly different from equalities as it deals with the less than, \(<\), or greater than, \(>\), symbols. While absolute value equations like \(|x+2|=9\) focus on equal aspects, expanding our knowledge into inequalities helps us understand all scenarios where the absolute value expression might be less or greater than a given number.Absolute value inequalities take form such as \(|x+2|<9\) or \(|x+2|>9\). Solving such inequalities involves understanding that:

• For \(|x+2|<9\), we determine the range \(-9
• For \(|x+2|>9\), we solve for \(x+2>9\) and \(x+2<-9\) to determine the conditions under which the inequality holds true.

By mastering inequalities alongside equations, one gets a comprehensive grasp of how absolute values behave in different scenarios. It streamlines problem-solving processes and prepares you for more advanced topics in algebra.