Question

Solve each equation. \(7-3|4 x-7|=4\)

Short answer

Solutions are \(x = \frac{31}{12}\) and \(x = \frac{11}{12}\).

Step-by-step solution

Isolate the Absolute Value

First, add 3 to both sides of the equation to isolate the absolute value term: \(7 - 3|4x - 7| + 3 = 4 + 3\) which simplifies to \(10 = 3|4x - 7|\)

Divide Both Sides

Next, divide both sides by 3: \(\frac{10}{3} = |4x - 7|\)

Set Up Two Equations

Set up two separate equations to account for the positive and negative solutions to the absolute value equation: \(4x - 7 = \frac{10}{3}\) and \(4x - 7 = -\frac{10}{3}\)

Solve the First Equation

Solve for \(x\) in the first equation: \(4x - 7 = \frac{10}{3}\)Add 7 to both sides: \(4x = \frac{10}{3} + 7 = \frac{10}{3} + \frac{21}{3} = \frac{31}{3}\)Divide by 4: \(x = \frac{31}{3} \times \frac{1}{4} = \frac{31}{12}\)

Solve the Second Equation

Solve for \(x\) in the second equation: \(4x - 7 = -\frac{10}{3}\)Add 7 to both sides: \(4x = -\frac{10}{3} + 7 = -\frac{10}{3} + \frac{21}{3} = \frac{11}{3}\)Divide by 4: \(x = \frac{11}{3} \times \frac{1}{4} = \frac{11}{12}\)

Key concepts

solving absolute value equations

Solving absolute value equations can seem tricky at first, but once you understand the steps, it becomes much simpler. To start, recall that the absolute value of a number is its distance from zero, regardless of direction. This means an absolute value equation will have two possible solutions: one positive and one negative. In other words, when you solve for the variable, you need to consider both positive and negative outcomes.
If you're given an equation like \(7-3|4 x-7|=4\), your goal is to 'free' the absolute value expression so you can handle it more easily. Just follow the methodical steps of isolating the absolute value and setting up separate equations for the positive and negative scenarios. Each step brings you closer to the final solution.
Whether you’re dealing with linear equations, inequalities, or real-world scenarios, mastering absolute value equations will always come in handy.

isolating absolute value

The first critical step to solving any absolute value equation is isolating the absolute value expression. Think of it like peeling away layers to get to the core of the problem. Start by moving all other terms to one side of the equation.
Looking at the exercise \(7-3|4 x-7|=4\), our first goal is to get the absolute value term \(|4x - 7|\) by itself. To do this:

• Add 3 to both sides:
• \(7 - 3|4x - 7| + 3 = 4 + 3\)
• Which simplifies to:
• \(10 = 3|4x - 7|\)

Now that the absolute value term is isolated, the equation becomes simpler and ready for the next step:

• Divide both sides by 3:
• \(\frac{10}{3} = |4x - 7|\)

Isolating the absolute value makes it easier to split the problem into two separate equations. It’s an incremental yet crucial move toward solving for the variable.

positive and negative solutions

When solving absolute value equations, remember that for any absolute value expression |A|, it equals A or -A. After isolating the absolute value term, you set up two equations: one where the expression inside the absolute value equals the positive right-hand side, and another where it equals the negative.
From our isolated equation \(\frac{10}{3} = |4x - 7|\), we write two new equations:

• \(4x - 7 = \frac{10}{3}\)
• \(4x - 7 = -\frac{10}{3}\)

To solve the first equation:

• \4x - 7 = \frac{10}{3}\
• Add 7: \4x = \frac{10}{3} + 7 = \frac{10}{3} + \frac{21}{3} = \frac{31}{3}\
• Divide by 4: \x = \frac{31}{12}\

To solve the second equation:

• \4x - 7 = -\frac{10}{3}\
• Add 7: \4x = -\frac{10}{3} + 7 = -\frac{10}{3} + \frac{21}{3} = \frac{11}{3}\
• Divide by 4: \x = \frac{11}{12}\

Therefore, the solutions are \(x = \frac{31}{12}\) and \(x = \frac{11}{12}\). Always remember to identify both positive and negative solutions to ensure you're covering all bases.