What is an inequality?

An inequality is a mathematical statement that compares two expressions, indicating that one is less than, greater than, less than or equal to, or greater than or equal to another. For example, x < 5 means x is less than 5.

Solve the inequality: 2x - 3 < 7.

Add 3 to both sides: 2x < 10. Then divide both sides by 2: x < 5. The solution is x < 5.

If |x| < 3, what are the possible values of x?

The inequality |x| < 3 implies that x is between -3 and 3. Therefore, -3 < x < 3.

Solve the absolute value equation: |2x - 4| = 6.

This can be solved as two separate equations: 2x - 4 = 6 and 2x - 4 = -6. Solving these gives x = 5 and x = -1.

Graph the inequality: x² > 4.

First, factor the inequality: (x - 2)(x + 2) > 0. The critical points are x = -2 and x = 2. The solution is x < -2 or x > 2, which can be shaded on a number line.

What is the fundamental property of inequalities when multiplying or dividing by a negative number?

When you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. For example, if -2x > 4, dividing by -2 gives x < -2.

What are the steps to solve the inequality: 3(x - 1) ≥ 2x + 4?

First, distribute: 3x - 3 ≥ 2x + 4. Then, subtract 2x from both sides: x - 3 ≥ 4. Finally, add 3 to both sides: x ≥ 7.

Given the inequality |x + 1| ≥ 5, what are the possible values of x?

This inequality splits into two cases: x + 1 ≥ 5 or x + 1 ≤ -5. Solving these gives x ≥ 4 or x ≤ -6. Therefore, the solution is x ≤ -6 or x ≥ 4.

What is the absolute value definition?

The absolute value of a number x, denoted |x|, is the non-negative value of x without regard to its sign. For example, |5| = 5 and |-5| = 5.

If 2|x - 3| < 8, what is the range of x?

First, divide by 2: |x - 3| < 4. This leads to two inequalities: -4 < x - 3 < 4. Adding 3 to all parts gives -1 < x < 7.

Solve the inequality: 4 - 3x > 1.

Subtract 4 from both sides: -3x > -3. Dividing by -3 reverses the inequality: x < 1.

Evaluate the inequality: x² - 4x + 3 ≤ 0.

Factor to get (x - 1)(x - 3) ≤ 0. The critical points are x = 1 and x = 3. The solution set is 1 ≤ x ≤ 3.