Question

Solve \(2|3 x-1|+4>10\)

Short answer

x < -\frac{2}{3} or x > \frac{4}{3}

Step-by-step solution

Isolate the Absolute Value Expression

First, isolate the absolute value expression by subtracting 4 from both sides of the inequality: \[ 2|3x - 1| + 4 > 10 \] Subtract 4: \[ 2|3x - 1| > 6 \]

Divide Both Sides by 2

Next, divide both sides of the inequality by 2 to further isolate the absolute value term: \[ |3x - 1| > 3 \]

Set Up the Two Inequalities

The absolute value expression \(|3x - 1| > 3\) means two separate inequalities: \[ 3x - 1 > 3 \] and \[ 3x - 1 < -3 \]

Solve Each Inequality

Solve the first inequality: \[ 3x - 1 > 3 \] Add 1 to both sides: \[ 3x > 4 \] Divide by 3: \[ x > \frac{4}{3} \] Now solve the second inequality: \[ 3x - 1 < -3 \] Add 1 to both sides: \[ 3x < -2 \] Divide by 3: \[ x < -\frac{2}{3} \]

Write the Final Solution

Combine the solutions from the two inequalities: \[ x < -\frac{2}{3} \] or \[ x > \frac{4}{3} \]

Key concepts

Absolute Value

Absolute value refers to the distance a number is from zero on a number line, regardless of direction. It's represented by vertical bars, like this: \(|x|\). The absolute value of any number is always non-negative. For example, \(|-3| = 3\) and \(|3| = 3\). This property is important in many areas of math, including when solving inequalities.

In the exercise, the absolute value expression is \(|3x-1|\). This expression varies depending on whether \(3x-1\) is positive or negative. Understanding this concept is crucial for breaking down problems involving absolute values.

Inequalities

Inequalities are expressions that show the relationship between two quantities where one is greater than, less than, or not equal to the other. Symbols like \(>\), \(<\), \(\leq\), and \(\geq\) are used to represent these relationships.

For example, in the inequality \(x > 2\), it means \(x\) is any number greater than 2. Inequalities can have a range of solutions rather than a single number.

In our exercise, we have the inequality \(2|3x-1| + 4 > 10\). Understanding how to manipulate and solve these is essential for finding the range of values that satisfy the inequality.

Isolate Variable

Isolating the variable means getting the variable alone on one side of the equation or inequality. This process usually involves performing inverse operations like addition, subtraction, multiplication, or division.

To isolate the absolute value expression in our exercise, we first subtract 4 from both sides:

\[2|3x - 1| + 4 > 10 \rightarrow 2|3x - 1| > 6\]
Next, divide by 2:

\[2|3x - 1| > 6 \rightarrow |3x - 1| > 3\]
These steps simplify the problem, making it easier to handle.

Remember, when isolating variables, always perform the same operation to both sides of the inequality to maintain balance.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations or inequalities to solve for a variable. This often includes using basic operations and properties of numbers.

In our exercise, after isolating the absolute value, we split it into two separate inequalities:

\[|3x - 1| > 3 \rightarrow 3x - 1 > 3 \ \text{or} \ 3x - 1 < -3\]
We then solve each inequality separately:

\[3x - 1 > 3 \: \text{Add 1 to both sides:} \ 3x > 4 \: \text{Divide by 3:} \ x > \frac{4}{3}\]
\[3x - 1 < -3 \: \text{Add 1 to both sides:} \ 3x < -2 \: \text{Divide by 3:} \ x < -\frac{2}{3}\]
This process allows us to find all possible values for \(x\) that satisfy the original inequality. Understanding and applying algebraic manipulation is key to solving such problems efficiently.