Question

Solve the equation. $$|x+1|=6$$

Short answer

The solutions to the absolute value equation \(|x+1|=6 \) are \(x=5\) and \(x=-7\).

Step-by-step solution

Understand the Absolute Value Function

Understand that the absolute value function \(|x+a|\) equals \(b\) has two solutions: one when \((x+a)=b\) and one when \((x+a)=-b\).

Set Up Two Equations

Therefore, for the given equation \(|x+1|=6\), we can set up two equations: \((x+1)=6\) and \((x+1)=-6\).

Solve the Two Equations

Solve both equations. For the first equation \((x+1)=6\), we find that \(x=6-1=5\). For the second equation \((x+1)=-6\), we find that \(x=-6-1=-7\). Therefore, the solutions to the given absolute value equation are \(x=5\) and \(x=-7\).

Key concepts

Solving Absolute Value Equations

To solve an absolute value equation, such as \(|x+1|=6\), it is essential to understand that the absolute value function represents the distance of a number from zero, regardless of direction on the number line.

In mathematical terms, \(|a| = b\) if and only if \(a = b\) or \(a = -b\). This means that when you see an equation like \(|x+1|=6\), you're looking for two things: when \(x+1\) itself equals 6 and when it equals -6, because both situations will satisfy the original absolute value equation. This split transforms one equation into two separate linear equations to solve.

• Equation 1: \(x + 1 = 6\), solve for \(x\) to get \(x = 5\).
• Equation 2: \(x + 1 = -6\), solve for \(x\) to get \(x = -7\).

The solutions to the equation \(|x+1|=6\) are the numbers 5 and -7.

Algebraic Equations

Algebraic equations are like puzzles that require finding values for variables that make an equation true. They are the very building blocks of algebra and can range from simple linear equations to more complex polynomial equations.

The goal of solving these equations is to isolate the variable on one side to find its value. This often involves performing operations such as addition, subtraction, multiplication, or division on both sides of the equation. When the equation contains an absolute value, like \(|x+1|=6\), it represents a pivotal step because it introduces the necessity for considering two separate cases, due to the nature of the absolute value function being always non-negative.

Additional techniques may come into play, especially as equations become more intricate, such as factoring, using the quadratic formula, or applying properties of exponents.

Absolute Value Function

The absolute value function is graphically represented as a 'V' shape on the coordinate plane, where the point at the bottom of the 'V', known as the vertex, occurs at the origin unless the function is shifted.

In the context of the function \(|x+a|\), the graph shifts \(a\) units to the left if \(a\) is positive or to the right if \(a\) is negative. This mirrors the way absolute value equations are solved algebraically by considering both positive and negative solutions.

A function like \(|x+1|\), being non-negative, implies that its graph will remain at or above the x-axis. This helps to visualize why absolute value equations produce two solutions: one on each side of the vertex, correlating to the positive and negative scenarios that are explored during algebraic solution processes.