02.05 Inequalities Notes

Steps for graphing inequalities

1. Rearrange the inequality into slope-intercept form. 2. Plot the y-intercept and use the slope to find a second point. 3. Connect the points with a dashed or solid line. 4. Shade above or below the line. Test the origin (0, 0) to make sure the correct side is shaded.

Equation Example

2x equals 4 is an example of an equation. Here, both sides are equal or balanced. Notice, there is only 1 number x that can make this equation true. x must equal 2 because 2 times 2 equals 4.

Closed Circle

A closed circle is used for ≥ (greater than or equal to) or ≤ (less than or equal to), because the number is part of the solution.

Linear inequality vs. Linear equation

A linear inequality is graphed the same way a linear equation is graphed with an addition of three things to consider. Good news though, you know all three of these things already! When solving for y by rearranging the inequality into slope-intercept form, remember to flip the inequality symbol when multiplying or dividing both sides by a negative number. "Open or closed?" turns to "solid or dashed line?" "Left or right?" turns to "shade above or below the line?"

Two variable linear inequalities

A system of linear inequalities in two variables consists of at least two linear inequalities in the same variables. The solution of a linear inequality is the ordered pair that is a solution to all inequalities in the system and the graph of the linear inequality is the graph of all solutions of the system.

Open Circle

An open circle is used for > (greater than) or < (less than), because the number is not part of the solution

Question 2: Which graph shows the solutions for y is less than or equal to 2 times x minus 3? A. A coordinate plane shows a solid line rising from left to right. The shading is to the right of this line. B. A coordinate plane shows a dashed line rising from left to right. The shading is to the right of this line. C. A coordinate plane shows a solid line rising from left to right. The shading is to the left of this line. D. A coordinate plane shows a dashed line rising from left to right. The shading is to the right of this line.

Answer is: C ....because the "less than or equal to" symbol tells you to use a solid line and shade below the boundary line.

Question 1: Which graph shows the solutions for y is greater than negative one-half times x, plus 2? A. A coordinate plane shows a solid line falling from left to right. The shading is below this line. B. A coordinate plane shows a dashed line falling from left to right. The shading is below this line. C. A coordinate plane shows a solid line falling from left to right. The shading is above this line. D. A coordinate plane shows a dashed line falling from left to right. The shading is above this line.

Answer is: D ...because the "greater than" symbol tells you to use a dashed line and to shade above the boundary line.

Question 3: Which graph shows the solutions for 5 times x plus 2 times y is less than or equal to 10? A. A coordinate plane shows a solid line falling from left to right. The shading is to the right of this line. B. A coordinate plane shows a dashed line falling from left to right. The shading is to the right of this line. C. A coordinate plane shows a dashed line falling from left to right. The shading is to the left of this line. D. A coordinate plane shows a solid line falling from left to right. The shading is to the left of this line.

Answer is: either A or D

Shade to the Left or Right?

It is very important to always read the inequality out loud and always start with the variable. Remember, the numbers on the number line become greater than to the right and less than to the left. Tip: When reading an inequality, the large opening is the greater than side, and the small point is the less than side.

Inequality

The value of the variable is not equal to one number (like in equations), but instead may be greater than or less than a number.

Extra Note:

When multiplying or dividing both sides of an inequality by a negative number, you must flip the inequality symbol.