Problem 91 - Algebra and Trigonometry

Question

In Exercises 59–94, solve each absolute value inequality. $$ 12<\left|-2 x+\frac{6}{7}\right|+\frac{3}{7} $$

Step-by-Step Solution

Answer

The solutions are \(x < -\frac{53}{14}\) and \(x > \frac{53}{14}\).

Step 1: Isolate the Absolute Value

The first step is to isolate the absolute value on one side of the inequality. In order to do that, you need to subtract \(\frac{3}{7}\) from both sides. The inequality will then become: \(12 - \frac{3}{7} < |-2x + \frac{6}{7}| \).

Step 2: Simplify the Left Side

After subtracting \(\frac{3}{7}\) from \(12\), simplify the left side of the inequality. It becomes: \(11\frac{4}{7} < |-2x + \frac{6}{7}|\).

Step 3: Solve the Two Resulting Inequalities

Since we are dealing with an absolute value inequality, this means that the expression inside the absolute value brackets can be either positive or negative and still satisfy the inequality. This gives us two inequalities to solve: \(11\frac{4}{7} < -2x + \frac{6}{7}\) and \(11\frac{4}{7} > 2x - \frac{6}{7}\).

Step 4: Solve for 'x'

Next, solve each inequality for 'x'. This involves subtracting \(\frac{6}{7}\) from each side, then dividing by \(-2\). The solutions are \(x < -\frac{53}{14}\) and \(x > \frac{53}{14}\).

Key Concepts

Inequality Solving

Inequality solving is akin to solving an equation, but instead of an equals sign, there can be signs such as "<" or ">". It involves finding the set of values that make the inequality true. This process can involve a series of steps that are critical for reaching the solution.

One of the key steps in solving inequalities is to understand that the inequality sign flips when you multiply or divide both sides by a negative number. This is crucial when solving absolute value inequalities.

• Identify the type of inequality and determine the relationships between the variables.
• Carefully manipulate the inequality, keeping track of operations that could alter the inequality sign, such as multiplying by negatives.

In the context of absolute value inequalities, you deal with two scenarios due to the nature of absolute values being able to represent both negative and positive values.

Isolation of Absolute Value

Isolating the absolute value is an essential initial step in solving absolute value inequalities. It involves ensuring that the absolute value expression is by itself on one side of the inequality.

For instance, if you have an inequality like:\[12<\left|-2 x+\frac{6}{7}\right|+\frac{3}{7}\]The strategy is to remove any additional terms that are added or subtracted to the absolute value. This can be achieved by performing inverse operations, like subtracting or adding the same value on both sides.

• The goal is to have the form \ \(\left|expression\right| < constant \ \), which allows for easier interpretation of the inequality.
• This helps to set up the next steps where you split and solve the resulting inequalities.

Ensuring the absolute value is isolated makes the next phase of solving much more straightforward.

Simplifying Inequalities

Once the absolute value is isolated, the next step is simplifying the inequality. This involves performing arithmetic operations to reduce the expression to a more manageable form.

This can involve combining like terms or converting fractions so they can be easily combined. For the exercise in question, after isolating the absolute value expression, the left side needed to be simplified:\[12 - \frac{3}{7} < \left|-2x + \frac{6}{7}\right|\]Simplifying gives:\[11\frac{4}{7} < \left|-2x + \frac{6}{7}\right|\]

• Ensure all arithmetic operations within the inequality are performed accurately.
• Keep track of these simplifications as they are critical for solving the inequality accurately.

This process makes it easier to set up the two cases in the absolute value inequality.

Solution of Inequalities

The solution of inequalities, especially those involving absolute values, boils down to solving two separate inequalities. This is because the expression inside the absolute value can either be positive or negative, yet still satisfy the inequality.

For the given inequality:\[11\frac{4}{7} < \left|-2x + \frac{6}{7}\right|\]This means you'll solve:

• \(11\frac{4}{7} < -2x + \frac{6}{7}\)
• \(11\frac{4}{7} > 2x - \frac{6}{7}\)

Both of these inequalities need to be solved for \(x\). Typically, you will:

• Isolate \(x\) in both inequalities by subtracting constants on both sides.
• Divide by the coefficient of the \(x\) term, remembering to flip the inequality sign if dividing by a negative.

This will yield two solution sets, which represent the values of \(x\) that satisfy the original absolute value inequality.