OBJECTIVES

1.4 Solving Inequalities

dUse < or > for to

write a true statement in a situation like 6 10.

Slide 1Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

a Determine whether a given number is a solution of aninequality.

b Graph an inequality on the number line.c Solve inequalities using the addition principle.d Solve inequalities using the multiplication principle.e Solve inequalities using the addition principle and the

multiplication principle together.

1.4 Solving Inequalities

aDetermine whether a given number is a solution of aninequality.

Slide 2Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

1.4 Solving Inequalities

aDetermine

whether a given number is a solution of an

inequality.

SOLUTION

Slide 3Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

A replacement that makes an inequality true is called a solution. The set of all solutions is called the solution set. When we have found the set of all solutions of an inequality, we say that we have solved the inequality.

EXAMPLE

1.4 Solving Inequalities

a Determine whether a given number is a solution of aninequality.

Slide 4Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

Determine whether 2 is a solution of x < 2.

EXAMPLE

1.4 Solving Inequalities

aDetermine whether a given number is a solution of aninequality.

Slide 5Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

Determine whether 6 is a solution of

1.4 Solving Inequalities

bGraph an inequality on the number line.

Slide 6Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

A graph of an inequality is a drawing that represents its solutions. An inequality in one variable can be graphed on the number line. An inequality in two variables can be graphed on the coordinate plane.

EXAMPLE

1.4 Solving Inequalities

b Graph an inequality on the number line.

Slide 7Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

The solutions are all those numbers less than 2. They are shown on the number line by shading all points to the left of 2. The open circle at 2 indicates that 2 is not part of the graph.

EXAMPLE

1.4 Solving Inequalities

b Graph an inequality on the number line.

Slide 8Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

The solutions are shown on the number line by shading the point for –3 and all points to the right of –3. The closed circle at –3 indicates that –3 is part of the graph.

EXAMPLE

1.4 Solving Inequalities

b Graph an inequality on the number line.

Slide 9Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

The inequality is read “–3 is less than or equal to x and x isless than 2,” or “x is greater than or equal to –3 and x is less than 2.” In order to be a solution of this inequality, a number must be a solution of both and x < 2. We can see from the graphs that the solution set consists of the numbers that overlap in the two solution sets in Examples 5 and 6.

EXAMPLE

1.4 Solving Inequalities

b Graph an inequality on the number line.

Slide 10Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

The open circle at 2 means that 2 is not part of the graph. The closed circle at –3 means that is part of the graph. The other solutions are shaded.

1.4 Solving Inequalities

c

Solve inequalities using the addition principle.

Slide 11Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

Any solution of one inequality is a solution of the other—they are equivalent.

1.4 Solving Inequalities

cSolve inequalities using the addition principle.

THE ADDITION PRINCIPLE FOR INEQUALITIES

Slide 12Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

For any real numbers a, b, and c:

In other words, when we add or subtract the same number on both sides of an inequality, the direction of the inequality symbol is not changed.

1.4 Solving Inequalities

cSolve inequalities using the addition principle.

Slide 13Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

As with equation solving, when solving inequalities, our goal is to isolate the variable on one side. Then it is easier to determine the solution set.

EXAMPLE

1.4 Solving Inequalities

c Solve inequalities using the addition principle.

Slide 14Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

1.4 Solving Inequalities

c Solve inequalities using the addition principle.

Slide 15Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

is read

A shorter notation for sets is called set-builder notation.

1.4 Solving Inequalities

dSolve inequalities using the multiplication principle.

THE MULTIPLICATION PRINCIPLE FOR INEQUALITIES

Slide 16Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

For any real numbers a and b, and any positive number c:

For any real numbers a and b, and any negative number c:

Similar statements hold for

1.4 Solving Inequalities

dSolve inequalities using the multiplication principle.

THE MULTIPLICATION PRINCIPLE FOR INEQUALITIES

Slide 17Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

In other words, when we multiply or divide by a positive number on both sides of an inequality, the direction of the inequality symbol stays the same. When we multiply or divide by a negative number on both sides of an inequality, the direction of the inequality symbol is reversed.

EXAMPLE

1.4 Solving Inequalities

d Solve inequalities using the multiplication principle.

Slide 18Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE

1.4 Solving Inequalities

d Solve inequalities using the multiplication principle.

Slide 19Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

1.4 Solving Inequalities

eSolve inequalities using the addition principle and themultiplication principle together.

Slide 20Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

Remember to reverse the inequality symbol when multiplying or dividing on both sides by a negative number.

EXAMPLE

1.4 Solving Inequalities

e Solve inequalities using the addition principle and themultiplication principle together.

Slide 21Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE

1.4 Solving Inequalities

eSolve inequalities using the addition principle and themultiplication principle together.

Slide 22Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE

1.4 Solving Inequalities

eSolve inequalities using the addition principle and themultiplication principle together.

Slide 23Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

First, we use the distributive law to remove parentheses. Next, we collect like terms and then use the addition and multiplication principles for inequalities to get an equivalent inequality with x alone on one side.

EXAMPLE

1.4 Solving Inequalities

eSolve inequalities using the addition principle and themultiplication principle together.

Slide 24Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE

1.4 Solving Inequalities

eSolve inequalities using the addition principle and themultiplication principle together.

Slide 25Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

The greatest number of decimal places in any one number is two. Multiplying by 100, which has two 0’s, will clear decimals. Then we proceed as before.

EXAMPLE

1.4 Solving Inequalities

eSolve inequalities using the addition principle and themultiplication principle together.

Slide 26Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE

1.4 Solving Inequalities

eSolve inequalities using the addition principle and themultiplication principle together.

Slide 27Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

The number 6 is the least common multiple of all the denominators. Thus we first multiply by 6 on both sides to clear the fractions.

EXAMPLE

1.4 Solving Inequalities

eSolve inequalities using the addition principle and themultiplication principle together.

Slide 28Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE

1.4 Solving Inequalities

eSolve inequalities using the addition principle and themultiplication principle together.

Slide 29Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.