Start Here

Linear Inequalities in One Variable

Basic ideas Statements that express the inequality of algebraic expressions are called inequalities.

Inequality Symbols

Basic ideas We know 4 < 5 but how do we compare -4 and -5 -1 5 4 3 2 1 -5 -4 -3 -2

-1 5 4 3 2 1 -5 -4 -3 -2

-1 5 4 3 2 1 -5 -4 -3 -2

-1 5 4 3 2 1 -5 -4 -3 -2

-1 5 4 3 2 1 -5 -4 -3 -2

-1 5 4 3 2 1 -5 -4 -3 -2

-1 5 4 3 2 1 -5 -4 -3 -2

-1 5 4 3 2 1 -5 -4 -3 -2

-1 5 4 3 2 1 -5 -4 -3 -2

-1 5 4 3 2 1 -5 -4 -3 -2

-1 5 4 3 2 1 -5 -4 -3 -2

-1 5 4 3 2 1 -5 -4 -3 -2

Graphing Inequalities The statement x < 3 means that x is a real number to the left of 3 on the number line.

Graphing Inequalities The statement x < 3 means that x is a real number to the left of 3 on the number line. The numbers 1.5, 0 and -2 are all to the left of 3 on the number line so the following are true.

Graphing Inequalities The statement x < 3 means that x is a real number to the left of 3 on the number line. The numbers 1.5, 0 and -2 are all to the left of 3 on the number line so the following are true. -1 5 4 3 2 1 -5 -4 -3 -2

Graphing Inequalities The statement x < 3 means that x is a real number to the left of 3 on the number line. The numbers 1.5, 0 and -2 are all to the left of 3 on the number line so the following are true. 1.5<3 -1 5 4 3 2 1 -5 -4 -3 -2

Graphing Inequalities The statement x < 3 means that x is a real number to the left of 3 on the number line. The numbers 1.5, 0 and -2 are all to the left of 3 on the number line so the following are true. 1.5<3 -1 5 4 3 2 1 -5 -4 -3 -2

Graphing Inequalities The statement x < 3 means that x is a real number to the left of 3 on the number line. The numbers 1.5, 0 and -2 are all to the left of 3 on the number line so the following are true. 1.5<3 0<3 -1 5 4 3 2 1 -5 -4 -3 -2

Graphing Inequalities The statement x < 3 means that x is a real number to the left of 3 on the number line. The numbers 1.5, 0 and -2 are all to the left of 3 on the number line so the following are true. 1.5<3 0<3 -1 5 4 3 2 1 -5 -4 -3 -2

Graphing Inequalities The statement x < 3 means that x is a real number to the left of 3 on the number line. The numbers 1.5, 0 and -2 are all to the left of 3 on the number line so the following are true. 1.5<3 0<3 -2<3 -1 5 4 3 2 1 -5 -4 -3 -2

Graphing Inequalities The statement x < 3 means that x is a real number to the left of 3 on the number line. The numbers 1.5, 0 and -2 are all to the left of 3 on the number line so the following are true. 1.5<3 0<3 -2<3 -1 5 4 3 2 1 -5 -4 -3 -2

Graphing Inequalities The set of all numbers that give a true statement when used as a replacement for x is called the solution set to the inequality. -1 5 4 3 2 1 -5 -4 -3 -2

Graphing Inequalities The set of all numbers that give a true statement when used as a replacement for x is called the solution set to the inequality. The solution set to x< 3 is all the real numbers to the left of 3 on a number line. -1 5 4 3 2 1 -5 -4 -3 -2

Graphing Inequalities The set of all numbers that give a true statement when used as a replacement for x is called the solution set to the inequality. The solution set to x< 3 is all the real numbers to the left of 3 on a number line. -1 5 4 3 2 1 -5 -4 -3 -2

Graphing Inequalities The set of all numbers that give a true statement when used as a replacement for x is called the solution set to the inequality. The solution set to x< 3 is all the real numbers to the left of 3 on a number line. -1 5 4 3 2 1 -5 -4 -3 -2

Open Interval

Open Interval (a

Open Interval (a

Open Interval (a

Open Interval

Open Interval (a )b

Open Interval (a )b

Open Interval (a )b

Open Interval

Open Interval )b

Open Interval )b

Open Interval )b

Half-open Interval

Half-open Interval [a

Half-open Interval [a

Half-open Interval [a

Half-opened Interval

Half-opened Interval (a

Half-opened Interval (a ]b

Half-opened Interval (a ]b

Half-opened Interval (a ]b

Half-opened Interval

Half-opened Interval [a

Half-opened Interval [a )b

Half-opened Interval [a )b

Half-opened Interval [a )b

Half-open Interval

Half-open Interval ]b

Half-open Interval ]b

Half-open Interval ]b

Closed Interval

Closed Interval [a

Closed Interval [a ]b

Closed Interval [a ]b

Closed Interval [a ]b

All real numbers

All real numbers

All real numbers

Interval Notation -1 5 4 3 2 1 -5 -4 -3 -2

Interval Notation -1 5 4 3 2 1 -5 -4 -3 -2

Interval Notation ) -1 5 4 3 2 1 -5 -4 -3 -2

Interval Notation ) -1 5 4 3 2 1 -5 -4 -3 -2

Interval Notation ) -1 5 4 3 2 1 -5 -4 -3 -2

Interval Notation ) -1 5 4 3 2 1 -5 -4 -3 -2

Interval Notation -1 5 4 3 2 1 -5 -4 -3 -2

Interval Notation -1 5 4 3 2 1 -5 -4 -3 -2

Interval Notation -1 5 4 3 2 1 -5 -4 -3 -2

Interval Notation [ -1 5 4 3 2 1 -5 -4 -3 -2

Interval Notation [ -1 5 4 3 2 1 -5 -4 -3 -2

Interval Notation [ -1 5 4 3 2 1 -5 -4 -3 -2

Interval Notation [ -1 5 4 3 2 1 -5 -4 -3 -2

-1 5 4 3 2 1 -5 -4 -3 -2

( -1 5 4 3 2 1 -5 -4 -3 -2

( -1 5 4 3 2 1 -5 -4 -3 -2

( -1 5 4 3 2 1 -5 -4 -3 -2

( -1 5 4 3 2 1 -5 -4 -3 -2

-1 5 4 3 2 1 -5 -4 -3 -2

] -1 5 4 3 2 1 -5 -4 -3 -2

] -1 5 4 3 2 1 -5 -4 -3 -2

] -1 5 4 3 2 1 -5 -4 -3 -2

] -1 5 4 3 2 1 -5 -4 -3 -2

A linear inequality in one variable x is any inequality of the form Ax + b < 0 where a and b are real numbers, with

A linear inequality in one variable x is any inequality of the form Ax + b < 0 where a and b are real numbers, with

A linear inequality in one variable x is any inequality of the form Ax + b < 0 where a and b are real numbers, with

A linear inequality in one variable x is any inequality of the form Ax + b < 0 where a and b are real numbers, with

A linear inequality in one variable x is any inequality of the form Ax + b < 0 where a and b are real numbers, with

Properties of Inequality

Properties of Inequality

Properties of Inequality

Properties of Inequality

Properties of Inequality Each side of an inequality may be multiplied by or divided by the same non-zero number without changing the solution set. If the inequality is multiplied by or divided by a negative number, the inequality symbol is reversed.

Properties of Inequality Each side of an inequality may be multiplied by or divided by the same non-zero number without changing the solution set. If the inequality is multiplied by or divided by a negative number, the inequality symbol is reversed.

Equaivalent Inequalities Equivalent inequalities are inequalities with the same solution set.

Solve x - 7 = -12

Solve x - 7 = -12

Solve x - 7 = -12

Solve x - 7 = -12

Solve x - 7 = -12

Solve x - 7 = -12 -1 5 4 3 2 1 -5 -4 -3 -2

Solve x - 7 = -12 -1 5 4 3 2 1 -5 -4 -3 -2

14 18 16 10 12

[ 14 18 16 10 12

[ 14 18 16 10 12

[ 14 18 16 10 12

-5 -6 -1 -2 -3 -4 -10 -9 -8 -7

] -5 -6 -1 -2 -3 -4 -10 -9 -8 -7

] -5 -6 -1 -2 -3 -4 -10 -9 -8 -7

] -5 -6 -1 -2 -3 -4 -10 -9 -8 -7

-5 -6 -1 -2 -3 -4 -10 -9 -8 -7

] -5 -6 -1 -2 -3 -4 -10 -9 -8 -7

] -5 -6 -1 -2 -3 -4 -10 -9 -8 -7

] -5 -6 -1 -2 -3 -4 -10 -9 -8 -7

-3 -4 2 1 -1 -2 -8 -7 -6 -5

( -3 -4 2 1 -1 -2 -8 -7 -6 -5

( -3 -4 2 1 -1 -2 -8 -7 -6 -5

( -3 -4 2 1 -1 -2 -8 -7 -6 -5

-1 5 4 3 2 1 -5 -4 -3 -2

[ -1 5 4 3 2 1 -5 -4 -3 -2

[ ] -1 5 4 3 2 1 -5 -4 -3 -2

[ ] -1 5 4 3 2 1 -5 -4 -3 -2

A rental company charges $15 to rent a chain saw, plus $2 per hour.

A rental company charges $15 to rent a chain saw, plus $2 per hour A rental company charges $15 to rent a chain saw, plus $2 per hour. Al Ghandi can spend no more than $35 to clear some logs from his yard.

A rental company charges $15 to rent a chain saw, plus $2 per hour A rental company charges $15 to rent a chain saw, plus $2 per hour. Al Ghandi can spend no more than $35 to clear some logs from his yard. What is the maximum amount of time he can use the rented saw?

A rental company charges $15 to rent a chain saw, plus $2 per hour A rental company charges $15 to rent a chain saw, plus $2 per hour. Al Ghandi can spend no more than $35 to clear some logs from his yard. What is the maximum amount of time he can use the rented saw?

Martha has scores of 88, 86, and 90 on here three algebra scores.

Martha has scores of 88, 86, and 90 on here three algebra scores Martha has scores of 88, 86, and 90 on here three algebra scores. An average score of at least 90 will earn an A in the class.

Martha has scores of 88, 86, and 90 on here three algebra scores Martha has scores of 88, 86, and 90 on here three algebra scores. An average score of at least 90 will earn an A in the class. What possible scores on her fourth test will earn her an A average.

Martha has scores of 88, 86, and 90 on here three algebra scores Martha has scores of 88, 86, and 90 on here three algebra scores. An average score of at least 90 will earn an A in the class. What possible scores on her fourth test will earn her an A average.

Martha has scores of 88, 86, and 90 on here three algebra scores Martha has scores of 88, 86, and 90 on here three algebra scores. An average score of at least 90 will earn an A in the class. What possible scores on her fourth test will earn her an A average.

Martha has scores of 88, 86, and 90 on here three algebra scores Martha has scores of 88, 86, and 90 on here three algebra scores. An average score of at least 90 will earn an A in the class. What possible scores on her fourth test will earn her an A average.

Martha has scores of 88, 86, and 90 on here three algebra scores Martha has scores of 88, 86, and 90 on here three algebra scores. An average score of at least 90 will earn an A in the class. What possible scores on her fourth test will earn her an A average.

Martha has scores of 88, 86, and 90 on here three algebra scores Martha has scores of 88, 86, and 90 on here three algebra scores. An average score of at least 90 will earn an A in the class. What possible scores on her fourth test will earn her an A average.

Martha has scores of 88, 86, and 90 on here three algebra scores Martha has scores of 88, 86, and 90 on here three algebra scores. An average score of at least 90 will earn an A in the class. What possible scores on her fourth test will earn her an A average.

L 1.5 # 1- 66