What is a linear inequality?

A linear inequality is an inequality that involves a linear expression, comparing two values using inequality symbols such as <, >, ≤, or ≥. For example, 2x + 3 < 7 is a linear inequality.

What is the addition rule of linear inequalities?

The addition rule states that if you add the same number to both sides of an inequality, the inequality remains unchanged. For example, if x > y, then x + a > y + a.

What happens to the inequality sign when you multiply or divide by a negative number?

When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. For example, if -3x < 9, dividing by -3 gives x > -3.

Solve the inequality: 5x - 2 ≥ 3. 

Hint: Isolate x on one side.

First, add 2 to both sides: 5x ≥ 5. Then, divide by 5: x ≥ 1. Thus, the solution is x ≥ 1.

What is the graphical representation of the inequality x < -2?

The graphical representation of x < -2 on a number line shows an open dot at -2 and a line extending to the left, indicating all numbers less than -2.

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

Hint: Break it into two separate inequalities.

First, break it into two parts: -3 < 2x + 1 and 2x + 1 ≤ 5. For the first part, subtract 1: -4 < 2x, then divide by 2: -2 < x. For the second part, subtract 1: 2x ≤ 4, then divide by 2: x ≤ 2. The combined solution is -2 < x ≤ 2.

What is the multiplication rule of linear inequalities?

The multiplication rule states that multiplying both sides of an inequality by a positive number does not change the inequality, while multiplying by a negative number reverses it.

If x + 4 < 10, what is the solution for x? 

Hint: Isolate x.

Subtract 4 from both sides: x < 6. Thus, the solution is x < 6.

What does it mean if a solution to an inequality is represented with a solid dot?

A solid dot indicates that the value is included in the solution set, which applies for inequalities like x ≤ 3.

Solve the inequality: 3(x - 2) > 6. 

Hint: Distribute and isolate x.

Distributing gives 3x - 6 > 6. Adding 6 to both sides gives 3x > 12. Dividing by 3 yields x > 4.

What is the difference between strict and non-strict inequalities?

Strict inequalities (<, >) do not include the endpoint, while non-strict inequalities (≤, ≥) do include the endpoint.

Graph the inequality 2y - x ≤ 4. 

Hint: First, convert to slope-intercept form.

Rearranging gives y ≤ (1/2)x + 2. The graph will have a solid line with shading below it, indicating all points where y is less than or equal to (1/2)x + 2.

How do you represent the inequality x ≥ 0 on a number line?

You will place a solid dot at 0 to indicate that 0 is included, and shade the line extending to the right to indicate all numbers greater than or equal to 0.

What is the solution to the inequality 4x + 1 < 13? 

Hint: Isolate x.

Subtract 1 from both sides: 4x < 12. Dividing by 4 gives x < 3. Thus, the solution is x < 3.

What is a system of linear inequalities?

A system of linear inequalities consists of two or more inequalities graphed on the same coordinate plane. The solution is the overlapping shaded region that satisfies all inequalities.