Solve each equation or inequality.
$\mid 7x-3\mid >4$

Verified step by step guidance

1. Recognize that the inequality involves an absolute value expression: \(|7x - 3| > 4\). Recall that for any expression \(A\), the inequality \(|A| > c\) (where \(c > 0\)) means \(A > c\) or \(A < -c\).
2. Set up two separate inequalities based on the definition of absolute value inequalities: 1) \$7x - 3 > 4$ 2) \$7x - 3 < -4$
3. Solve the first inequality \$7x - 3 > 4\( by adding 3 to both sides: \)7x > 7$ Then divide both sides by 7: \(x > 1\)
4. Solve the second inequality \$7x - 3 < -4\( by adding 3 to both sides: \)7x < -1$ Then divide both sides by 7: \(x < -\frac{1}{7}\)
5. Combine the two solution sets to express the final solution: \(x < -\frac{1}{7}\) or \(x > 1\)

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve expressions where the distance of a number from zero is compared to another value. For |A| > B, the solution splits into two cases: A > B or A < -B, because the absolute value measures magnitude regardless of sign.

Solving Linear Inequalities

Solving linear inequalities requires isolating the variable on one side while maintaining the inequality's direction. When multiplying or dividing by a negative number, the inequality sign reverses. Solutions are often expressed as intervals or compound inequalities.

Compound Inequalities

Compound inequalities combine two inequalities connected by 'and' or 'or'. For absolute value inequalities like |7x - 3| > 4, the solution is a union of two inequalities, representing values that satisfy either condition, reflecting the distance concept.