Solve the system of inequalities by graphing calculator

Equations in which the unknown only appear to the first power are called linear equations. The general form of a linear equation in one variable is ax + b = 0, where x is the variable.

A linear inequality resembles an equation by replacing the equal sign with an inequality symbol. Generally, a range of values, rather than one specific value will be the solution to a linear inequality. Some of the examples of Linear Inequation are listed below.

• x < 7
• 5 - y ≥ 30
• x + 5 > 20
• Q ≠ 50

Steps for Linear Inequalities Graphing

The linear inequality graph intersects the coordinate plane into two parts by a borderline. In order to plot the graph of an inequality, we have to follow some basic steps:

• Firstly, check if the given linear inequality equation is arranged properly or not.
• If not rearrange the equation, where the variable will be on the left side and the rest of the expression on the right side of the inequality symbol
• In this step, put the x values in the equation and plot the graph.
• Retain to draw a solid line for y≤ or y≥ and a dashed line for y< or y>.
• Finally, shade the line as per inequalities, like above the line for a “greater than” (y> or y≥) and below the line for a “less than” (y< or y≤).

Steps for Solving One Variable Linear Equations and Inequations

To solve the linear equation or inequality having only one variable, the steps provided below are followed to balance the equation:

• Firstly, add or subtract like terms
• Isolate the variable
• Reverse or eliminate the terms
• Check the answer.

Example:

Find inequality for 6x + 4 > 28

Solution:

Put the variable on the LHS and reverse all the other terms or the coefficient of x to the RHS.

6x > 28 - 4

6x > 24

 ⇒ x > 4

Hence, the value of x is greater than 4.

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To graph a linear inequality in two variables (say, x and y ), first get y alone on one side. Then consider the related equation obtained by changing the inequality sign to an equality sign. The graph of this equation is a line.

If the inequality is strict ( < or > ), graph a dashed line. If the inequality is not strict ( ≤ or ≥ ), graph a solid line.

Finally, pick one point that is not on either line ( ( 0 , 0 ) is usually the easiest) and decide whether these coordinates satisfy the inequality or not. If they do, shade the half-plane containing that point. If they don't, shade the other half-plane.

Graph each of the inequalities in the system in a similar way. The solution of the system of inequalities is the intersection region of all the solutions in the system.

Example 1:

Solve the system of inequalities by graphing:

y ≤ x − 2 y > − 3 x + 5

First, graph the inequality y ≤ x − 2 . The related equation is y = x − 2 .

Since the inequality is ≤ , not a strict one, the border line is solid.

Graph the straight line.

Consider a point that is not on the line - say, ( 0 , 0 ) - and substitute in the inequality y ≤ x − 2 .

0 ≤ 0 − 2 0 ≤ − 2

This is false. So, the solution does not contain the point ( 0 , 0 ) . Shade the lower half of the line.

Similarly, draw a dashed line for the related equation of the second inequality y > − 3 x + 5 which has a strict inequality. The point ( 0 , 0 ) does not satisfy the inequality, so shade the half that does not contain the point ( 0 , 0 ) .

The solution of the system of inequalities is the intersection region of the solutions of the two inequalities.

Example 2:

Solve the system of inequalities by graphing:

2 x + 3 y ≥ 12 8 x − 4 y > 1 x < 4

Rewrite the first two inequalities with y alone on one side.

3 y ≥ − 2 x + 12 y ≥ − 2 3 x + 4 − 4 y > − 8 x + 1 y < 2 x − 1 4

Now, graph the inequality y ≥ − 2 3 x + 4 . The related equation is y = − 2 3 x + 4 .

Since the inequality is ≥ , not a strict one, the border line is solid.

Graph the straight line.

Consider a point that is not on the line - say, ( 0 , 0 ) - and substitute in the inequality.

0 ≥ − 2 3 ( 0 ) + 4 0 ≥ 4

This is false. So, the solution does not contain the point ( 0 , 0 ) . Shade upper half of the line.

Similarly, draw a dashed line of related equation of the second inequality y < 2 x − 1 4 which has a strict inequality. The point ( 0 , 0 ) does not satisfy the inequality, so shade the half that does not contain the point ( 0 , 0 ) .

Draw a dashed vertical line x = 4 which is the related equation of the third inequality.

Here point ( 0 , 0 ) satisfies the inequality, so shade the half that contains the point.

The solution of the system of inequalities is the intersection region of the solutions of the three inequalities.

Can a TI

What are the 3 steps you must do to solve a system of inequalities?