Solve each equation or inequality.

Verified step by step guidance

1. Start by isolating the absolute value expression on one side of the inequality. Add 4 to both sides to get: \(| -5x + 7 | < -6 + 4\).
2. Simplify the right side of the inequality: \(| -5x + 7 | < -2\).
3. Recall that the absolute value of any expression is always greater than or equal to zero, so it can never be less than a negative number.
4. Since \(| -5x + 7 |\) cannot be less than \(-2\), there are no values of \(x\) that satisfy this inequality.
5. Therefore, the solution set is the empty set, meaning no solution exists for this inequality.

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve expressions where the absolute value of a variable expression is compared to a number. To solve them, consider the definition of absolute value as distance from zero, leading to two cases: one positive and one negative. For example, |A| < B means -B < A < B.

Isolating the Absolute Value Expression

Before solving an absolute value inequality, isolate the absolute value term on one side of the inequality. This often involves adding or subtracting constants and dividing by coefficients. Proper isolation is crucial to correctly apply the definition and split into cases.

Solving Linear Inequalities

After splitting the absolute value inequality into two linear inequalities, solve each separately using standard techniques. This includes adding, subtracting, multiplying, or dividing both sides by constants, remembering to reverse inequality signs when multiplying or dividing by negative numbers.