Question

Solve: \(|3 x+7|-3=5\)

Short answer

x=\frac{1}{3} and x=-5

Step-by-step solution

- Isolate the absolute value expression

Start by isolating the absolute value expression on one side of the equation. Add 3 to both sides:\(|3x+7|-3+3=5+3\)This simplifies to:\(|3x+7|=8\)

- Set up the two cases for the absolute value

The equation \(|3x+7|=8\) implies two cases:Case 1: \(3x+7=8\)Case 2: \(3x+7=-8\)

- Solve Case 1

Solve for \(x\) in the first case:\(3x+7=8\)Subtract 7 from both sides:\(3x=1\)Divide both sides by 3:\(x=\frac{1}{3}\)

- Solve Case 2

Solve for \(x\) in the second case:\(3x+7=-8\)Subtract 7 from both sides:\(3x=-15\)Divide both sides by 3:\(x=-5\)

- Combine the solutions

The solutions from both cases are:\(x=\frac{1}{3}\) and \(x=-5\)

Key concepts

absolute value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. It's always a non-negative number.
For instance, the absolute value of both 5 and -5 is 5 because both are 5 units away from zero. In algebraic terms, this is written as \(|x|\). When solving absolute value equations, such as \(|3x + 7| - 3 = 5\), it's essential to isolate the absolute value term first.
This makes it easier to set up two scenarios representing the positive and negative outcomes of the equation inside the absolute value.

algebraic equation

An algebraic equation includes variables, coefficients, and constants combined through algebraic operations (addition, subtraction, multiplication, and division).
The goal when solving such equations is to find the value of the variable that makes the equation true.
For example, in \(3x + 7 = 8\), you're solving for x. This means manipulating the equation to isolate x on one side. Solving algebraic equations often involves steps like adding, subtracting, multiplying, or dividing both sides of the equation.

linear equation

A linear equation is an equation where the variable is raised to the power of one. It has the general form \(ax + b = c\).
These equations graph as straight lines. In the context of our absolute value problem, once the absolute value is dealt with, each resulting equation is linear.
For example, solving \(3x + 7 = 8\) involves steps to isolate x by performing arithmetic operations. Linear equations are simpler to solve and serve as the foundation for more complex algebraic operations.

step-by-step solution

Breaking down problems into structured steps helps us understand and solve them efficiently.
Let's recap the steps to solve \(|3x + 7| - 3 = 5\).
First, we isolate the absolute value term: \(|3x + 7| = 8\).
Then, we consider two scenarios: \(3x + 7 = 8\) and \(3x + 7 = -8\).
Solving the first scenario for x, we subtract 7 and then divide by 3, getting \(x = \frac{1}{3}\).
Solving the second scenario, we again subtract 7 and divide by 3, yielding \(x = -5\).
This step-by-step approach leads us to the solutions: \(\frac{1}{3}\) and \(-5\).