## Graphing a Linear Inequality

Graphing a Linear Inequality

Step 1:-  Draw the graph of the straight line that is the boundary.

Make the line solid if the inequality involves ≤ or ≥; make the

line dashed if the inequality involves < or >.

Step 2:-  Choose a test point. Choose any point not on the line, and

substitute the coordinates of this point in the inequality.

the test point if it satisfies the original inequality; otherwise,

shade the region on the other side of the boundary line.

## Linear Inequality in Two Variables

Linear Inequality in Two Variables

An inequality that can be written as

Ax + By  <  C or Ax + By > C,

where A, B, and C are real numbers and A and B are not both 0, is a

linear inequality in two variables.

## Solving a Compound Inequality with or

Solving a Compound Inequality with or

Step 1:-  Solve each inequality in the compound inequality individually.

Step 2:-  Since the inequalities are joined with or, the solution set includes all numbers that satisfy either one of the two inequalities in Step 1 (the union of the solution sets).

## Solving a Compound Inequality with and

Solving a Compound Inequality with and

Step 1:-  Solve each inequality in the compound inequality individually.

Step 2:-  Since the inequalities are joined with and, the solution set of the compound inequality will include all numbers that satisfy both inequalities in Step 1 (the intersection of the solution sets).

## Multiplication Property of Inequality and Solving a Linear Inequality

Multiplication Property of Inequality

For all real numbers A, B, and C, with C Z 0,

(a) the inequalities

A <  B and AC  < BC

are equivalent if C  >  0;

(b) the inequalities

A < B and AC >BC

are equivalent if C < 0.

In words, each side of an inequality may be multiplied (or divided)

by a positive number without changing the direction of the inequality

symbol. Multiplying (or dividing) by a negative number requires that

we reverse the inequality symbol.

Solving a Linear Inequality

Step 1 Simplify each side separately. Use the distributive property to

clear parentheses and combine like terms as needed.

Step 2 Isolate the variable terms on one side. Use the addition property

of inequality to get all terms with variables on one side of

the inequality and all numbers on the other side.

Step 3 Isolate the variable. Use the multiplication property of

inequality to change the inequality to the form x <k or x > k.

## Solving a Linear Inequality

Solving a Linear Inequality

Step 1:-  Simplify each side separately. Use the distributive property to clear parentheses and combine like terms as needed.

Step 2:-  Isolate the variable terms on one side. Use the addition property of inequality to get all terms with variables on one side of

the inequality and all numbers on the other side.

Step 3:-  Isolate the variable. Use the multiplication property of inequality to change the inequality to the form x < k or x > k.

## Linear Inequality

Linear Inequality

A linear inequality in one variable can be written in the form

Ax +  B  <  C,

where A, B, and C are real numbers, with A ≠ 0.

For all real numbers A, B, and C, the inequalities

A < B and A + C < B + C

are equivalent.

In words, adding the same number to each side of an inequality does not change the solution  set.

Multiplication Property of Inequality

For all real numbers A, B, and C, with C Z 0,

(a) the inequalities

A < B and AC < BC

are equivalent if C > 0;

(b) the inequalities

A < B and AC > BC

are equivalent if C < 0.

In words, each side of an inequality may be multiplied (or divided)

by a positive number without changing the direction of the inequality

symbol. Multiplying (or dividing) by a negative number requires that

we reverse the inequality symbol.

## Use inequality symbols

Use inequality symbols

The statement 4 + 2 = 6 is

an equation; it states that two quantities are equal. The statement 4 ≠ 6

(read “4 is not equal to 6”) is an inequality, a statement that two quantities

are not equal. When two numbers are not equal, one must be less than the

other. The symbol <  means “is less than.” For example,

8 < 9, -6 <, 15, -6 < – 1, and 0 < .

The symbol >  means “is greater than.” For example,

12 > 5, 9 > -2, and .

In each case, the symbol “points” toward the smaller number.