What is an inequality?

An inequality is a mathematical statement that compares two expressions, showing that one is greater than, less than, greater than or equal to, or less than or equal to the other. Common symbols include >, <, ≥, and ≤.

If a < b and b < c, what can be concluded about a and c?

If a < b and b < c, it can be concluded that a < c by the transitive property of inequalities.

What is the formula for the absolute value of a number?

The absolute value of a number x is defined as: |x| = x if x ≥ 0; |x| = -x if x < 0. For example, |−3| = 3 and |4| = 4.

What is the product of two negative numbers?

The product of two negative numbers is always positive. For example, (−2) × (−3) = 6.

When solving inequalities, what must be done when multiplying or dividing by a negative number?

When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. For example, if a < b and c < 0, then ac > bc.

Solve the inequality: 3x - 5 < 7.

To solve for x, first add 5 to both sides: 3x < 12. Then, divide both sides by 3: x < 4.

What is the value of x in the inequality 5 - 2x ≥ 3?

First, subtract 5 from both sides: −2x ≥ −2. Then, divide by −2, reversing the inequality: x ≤ 1.

If x and y are real numbers such that x + y < 10 and x > 2, what can be concluded about y?

Rearranging the first inequality gives y < 10 - x. Since x > 2, substituting gives y < 10 - 2, so y < 8.

What is the inequality representation of a number being less than or equal to 5?

The inequality representation is x ≤ 5, which means x can take any value less than or equal to 5.

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

The inequality |x| ≤ 3 means that x is between -3 and 3, inclusive. Thus, the solution is -3 ≤ x ≤ 3.

Solve the compound inequality: -2 < 3x + 1 < 5.

First, break it into two inequalities: -2 < 3x + 1 and 3x + 1 < 5. Solving the first gives x > -1. The second gives x < 4/3. Thus, the solution is -1 < x < 4/3.

Consider the inequality: 2x - 3 < 4. What is the solution?

Add 3 to both sides to get 2x < 7. Then divide by 2, resulting in x < 3.5.

What can you infer from the inequality x² > 9?

To solve x² > 9, take the square root of both sides, giving x > 3 or x < -3. Thus, the solution set is x < -3 or x > 3.

If x + 4 ≤ 7 and x - 2 > 1, what is the range of possible values for x?

From x + 4 ≤ 7, we get x ≤ 3. From x - 2 > 1, we get x > 3. The inequalities are contradictory, meaning there is no value of x that satisfies both conditions.

For x² - 4x + 3 < 0, what are the roots of the equation?

First, factor the quadratic: (x - 1)(x - 3) < 0. The roots are x = 1 and x = 3. The solution to the inequality is 1 < x < 3.