Flash Cards: Linear Inequalities

An inequality is a mathematical statement that compares two expressions using symbols like '<', '>', '≤', or '≥'.

A linear inequality in one variable takes the form ax + b < 0, ax + b > 0, ax + b ≤ 0, or ax + b ≥ 0, where a and b are real numbers.

When both sides of an inequality are multiplied or divided by a negative number, the direction of the inequality sign reverses.

A strict inequality uses '<' or '>', indicating that the expressions are not equal. Example: x < 5.

To graph an inequality, shade the region that satisfies the inequality on a number line, using open or closed circles for strict or non-strict inequalities.

Dividing by 30 gives x < 20/3, or x can take values 0, 1, 2, ..., 6 if it's a natural number.

A double inequality is written in the form a < x < b, meaning x is greater than a and less than b.

A slack inequality uses '≤' or '≥', allowing equality to be part of the solution set.

Natural numbers are positive integers (1, 2, 3,...), while integers include negative numbers and zero (..., -2, -1, 0, 1, 2,...).

To have an average of 60 from three exams: 62 + 48 + x ≥ 180, leading to x ≥ 70.

Rearranging gives 2x < 4, so x < 2; solution set is x ∈ (-∞, 2) for real numbers.

On the number line, use an open circle at 3 and shade to the left to indicate x can take any value less than 3.

An algebraic solution involves rearranging terms using addition, subtraction, multiplication, or division, to isolate the variable.

Rearranging leads to x > -2; the solution set is (-2, ∞).

This is a linear inequality in two variables where a and b are not both zero.

A common mistake is not reversing the inequality sign when multiplying or dividing by a negative number.

Strict inequalities (e.g., x < 5) do not include the boundary value; slack inequalities (e.g., x ≤ 5) do include it.

A real-valued solution refers to the solutions of inequalities that can take any real number value.