Section 2.7

Solving Linear Inequalities

Learning objectives

Graph inequalities on a number line

Use the addition property of inequality

Use the multiplication property of inequality

Use both properties to solve inequalities

Solve problems modeled by inequalities

Vocabulary: inequality, at least, at most, no less than, no more

than, is less than, is greater than

Solving linear inequalities

Remember, from chapter 1:

< means “less than”.

> means “greater than”

≤ means “less than or equal to”. It can also mean “at most”

≥ means “greater than or equal to”. It can also mean “at least”

If I tell you that x is “at least” 12, that means x ≥ 12, or “x is

greater than or equal to 12; x is at least 12, if not more”

Graphing inequalities on a number line

An inequality is a statement that contains either <, >, ≥, or

≤. In other words, it does not contain an equal sign (=)

The statement x = 3 has one solution: 3

The statement x > 3 has many solutions: all real numbers

greater than 3

Because it would be a pain to list them all, we choose

instead to show the solution on a number line:

|----|----|----|----|----|----|----|----|----o----|----|----|----|--->

0 1 2 3 4 5 6 7

An open circle appears on x = 3, because 3 is NOT part

of the solution. All numbers greater than 3 are part

Example 1: graph: x ≥ -1

To graph the solutions for x greater than or equal to -1,

we put a closed circle on -1, since -1 is part of the

solution, then draw a line to the right of -1:

---|----|----|----|----|----|----|----|----|----|----|----|----|----|

Example 2: graph -1 > x

Keep in mind that -1 > x is the same as x < -1. When you

switch sides in an inequality, you must turn around the

sign.

An open circle is drawn at x = -1, since -1 is not part of

the solution. Then a line is drawn to the left

--|----|----|----|----|----|----|----|----|----|----|----|----|----|-

Example 3: graph -4 < x ≤ 2

The solution to this problem is all real numbers “greater

than -4” and “less than or equal to 2”.

On the number line, draw an open circle at -4 and a filled

in circle at 2. -4 is not part of the solution, but 2 is.

Then fill in the line between -4 and 2

--|----|----|----|----|----|----|----|----|----|----|----|----|----|-

Graph each inequality on a number line

x ≥ -5

--|----|----|----|----|----|----|----|----|----|----|----|----|----|-

Y < 4

--|----|----|----|----|----|----|----|----|----|----|----|----|----|-

-3/2 ≥ m

--|----|----|----|----|----|----|----|----|----|----|----|----|----|-

-5 < t ≤ 0

--|----|----|----|----|----|----|----|----|----|----|----|----|----|-

Using the addition property

Just like in equations, sometimes we need to add a

number to both sides of an inequality to solve it. The

addition property of inequality looks like the regular

property:

If a, b, and c are real numbers:

Then a > b and a + c > b + c

Are equivalent inequalities: they have the same solution

This is also true for subtracting the same number from

both sides of an inequality

Example 4: solve x + 4 ≤ 6

To solve for x, subtract 4 from both sides of the inequality

X + 4 – 4 ≤ 6 – 4

Therefore x ≤ 2. Graph the solution. Draw a closed o at

x = 2, and draw a line to the left.

--|----|----|----|----|----|----|----|----|----|----|----|----|----|-

Solve each inequality. Graph the solution

X + 7 ≤ 12

--|----|----|----|----|----|----|----|----|----|----|----|----|----|-

X – 10 > -3

--|----|----|----|----|----|----|----|----|----|----|----|----|----|-

-4z – 2 > -5z + 1

--|----|----|----|----|----|----|----|----|----|----|----|----|----|-

18 – 2x ≤ -3x + 24

--|----|----|----|----|----|----|----|----|----|----|----|----|----|-

Using the multiplication property

There is one important difference between the addition

property and the multiplication property.

In the addition property, you never have to switch the

direction of the signs if you add something to both sides

In the multiplication property, if you multiply both sides

by a negative number, you have the switch the direction of

the sign.

Ex. If 6 < 8, multiply both sides by 2, then 12 < 16? Sure.

But if we multiply both sides by -2, is -12 < -16?

The multiplication property of inequality is more

complicated than the addition

The multiplication property of inequality

If a, b, and c are real numbers, and c is positive,

Then: a < b and ac < bc are equivalent inequalities

(have the same solution)

But if c is negative, then a < b and ac > bc are equivalent

inequalities

Example 5: solve -2x ≤ -4

To get x by itself, we need to divide each side by -2:

(-2x)/-2 ≥ -4/-2

Notice we have to turn the sign around

Therefore x ≥ 2 is the solution to this inequality

--|----|----|----|----|----|----|----|----|----|----|----|----|----|-

Example 6: solve 2x < -4

In this case, to get x by itself, we divide both sides by 2

Therefore x < -2

We do not have to change the direction of the sign

--|----|----|----|----|----|----|----|----|----|----|----|----|----|-

Solve each inequality, graph the solution

-8 ≥ x/3

--|----|----|----|----|----|----|----|----|----|----|----|----|----|-

3x < 12

--|----|----|----|----|----|----|----|----|----|----|----|----|----|-

0 < y/8

--|----|----|----|----|----|----|----|----|----|----|----|----|----|-

-3/5y ≤ 9

--|----|----|----|----|----|----|----|----|----|----|----|----|----|-

Using both properties of inequality

This is done similarly to solving equations, using the same

five steps (shown on page 157): eliminate fractions;

distribute; combine like terms; isolate the variable; divide

for variable

The only difference: if you multiply or divide both sides by

a negative number, you have to reverse the inequality sign

There is one other difference: you have to graph your

answer on a number line:

--|----|----|----|----|----|----|----|----|----|----|----|----|----|-

Example 7: solve -4x + 7 ≥ -9

First, subtract 7 from both sides. Sign does not change

-4x + 7 – 7 ≥ -9 – 7.

-4x ≥ -16

Then, divide both sides by -4. The sign will change.

-4x/-4 ≤ -16/-4

Finally, x ≤ 4. Graph the solution set:

--|----|----|----|----|----|----|----|----|----|----|----|----|----|-

Example 8: solve -5x + 7 < 2(x – 3)

Distribute the right side:

-5x + 7 < 2x – 6

Subtract both sides by 7:

-5x + 7 – 7 < 2x – 6 – 7; -5x < 2x – 13

Then subtract 2x from both sides

-5x – 2x < 2x – 2x – 13; -7x < -13

Finally, divide both sides by -7. Reverse the sign

-7x/-7 > -13/-7; x > 13/7 or 1 6/7

--|----|----|----|----|----|----|----|----|----|----|----|----|----|-

Example 9: 2(x – 3) – 5 ≤ 3(x + 2) - 18

Distribute: 2x – 6 – 5 ≤ 3x + 6 – 18

Combine like terms: 2x – 11 ≤ 3x -12

Add 11 to both sides: 2x ≤ 3x – 1

Subtract 3x from both sides: -x ≤ -1

Finally, multiply both sides by -1. Reverse the sign. X ≥ 1

Graph the solution:

--|----|----|----|----|----|----|----|----|----|----|----|----|----|-

Using both properties, solve the following:

3(3x – 16) < 12(x – 2)

--|----|----|----|----|----|----|----|----|----|----|----|----|----|-

-18(z – 2) ≥ -21z + 24

--|----|----|----|----|----|----|----|----|----|----|----|----|----|-

(8/21)(x + 2) > (1/7)(x + 3)

--|----|----|----|----|----|----|----|----|----|----|----|----|----|-

Solving properties modeled by inequalities

Words such as “at least”, “at most”, “no more than” and

“no less than” suggest inequality problems

The sign < means “less than”

The sign ≤ means “less than or equal”. It also means “at

most” and “no more than”

The sign ≥ means “more than or equal”. It also means “at

least” and “no less than”

Example 10: 12 subtracted from 3 times a

number is less than 21

Let x be the unknown number. Translate the sentence:

3 times a number is: 3x. 12 subtracted from 3x is: 3x – 12

So if 3x – 12 is less than 21, then 3x – 12 < 21

Now, solve for x. Add 12 to each side:

3x – 12 + 12 < 21 + 12; 3x < 33

Now divide each side by 3. Do not switch sign

3x/3 < 33/3; The solution is: x < 11

Graph the solution:

--|----|----|----|----|----|----|----|----|----|----|----|----|----|-

Example 11: budgeting for a wedding

A couple have a $1000 wedding hall budget. The hall

charges a $100 rental plus $14 a person. What is the

greatest number of people that can attend?

Let x be the number of people that can attend.

The cost will be the $100 rental plus $14 per person,

which cannot exceed (is at most) $1000

The inequality will be: $100 + $14x ≤ $1000

Solve for x: $14x ≤ $900; x ≤ 900/14; x ≤ 64.3

65 people is too many: the cost would be over $1000

We can have, at most, 64 people at the wedding.

Solve the following:

8 more than twice a number is less than -12.

--|----|----|----|----|----|----|----|----|----|----|----|----|----|-

One side of a triangle is 6 times as long as another side,

and the third side is 8 inches long. The perimeter can be

no more than 106 inches. Find the lengths of the 2 sides.

Chapter 2 highlights

Equivalent equations are equations that have the same

solution

Ex. 2x = 6 and x + 4 = 7 are equivalent: both have x = 3

Addition property of equality: adding the same number to

both sides of an equation does not change the solution

9 + y = 3

9 – 9 + y = 3 – 9

So y = -6

Multiplication property of equality

Multiplying both sides or dividing both sides of an

equation by the same nonzero number does not change

the solution.

Ex. 2/3 a = 18; multiply both sides by 3/2

3/2 * 2/3 a = 3/2 * 18

Therefore a = 27

Solving linear equations – five steps

1) clear the equation of fractions

2) distribute to remove any parenthesis

3) simplify by combining like terms

4) isolate the variable terms to one side of equal sign

5) use division or multiplication to solve for the variable

Then check your work by substituting the solution into

the original problem

Solve: 5/6(-2x + 9) + 3 = 1/2

1) remove fractions by multiplying both sides by LCD of 6

6*5/6(-2x + 9) + 6*3 = 6*1/2

5(-2x + 9) + 18 = 3

2) distribute to remove parentheses

-10x + 45 + 18 = 3

3) combine like terms

-10x + 63 = 3

4) isolate the variable

-10x + 63 – 63 = 3 – 63; -10x = -60

5) use multiplication or division to solve for x

-10x/-10 = -60/-10; x = 6

Problem Solving

UNDERSTAND the problem

TRANSLATE it to an equation

SOLVE the equation

INTERPRET the result

The height of H volcano is twice that of K volcano. If the sum of both is 12,870 feet, find the height of each.

UNDERSTAND: the height of H is in terms of K: H=2K

TRANSLATE: if H + K = 12,870, then 2K + K = 12,870

SOLVE: 3K = 12870; K = 4290; H = 2K = 8580

INTERPRET: Volcano K is 4290 feet and H is 8580 feet

Formulas and problem solving

A formula is an equation that describes a known

relationship among quantities

Use the same steps as solving an equation

Solve the perimeter formula: P = 2l + 2w, for l

First, subtract 2w from both sides: P – 2w = 2l

Then, divide both sides by 2: (P – 2w)/2 = l

Percent and mixture problem solving

32% of what number is 36.8?

The problem says that 32% of x = 36.8

32% is equivalent to 0.32, so 0.32x = 36.8; x = 115

How many liters of 20% acid must be mixed with 50% acid to make 12 liters of a 30% solution?

Write each term as percent*volume, ex. 30%(12) =0.3(12)

So x liters of 20% + 12-x liters of 50% = 12 liters of 30%

0.20x + 0.50(12 – x) = 0.30(12)

0.20x + 6 – 0.50x = 3.6

-0.30x = -2.4; x = 8; 12 – x = 4

It takes 8 liters of 20% and 4 liters of 50% to make the 12 liters of the 30% solution.

Linear inequalities

Linear inequalities have <, >, ≤ or ≥ in them

Use the same 5-step procedure as solving equations

One difference: when multiplying or dividing both sides by a negative number, you must switch the direction of the sign

Solve: -3(x + 2) < -2 + 11

-3x – 6 < 9; now, add 6 to both sides

-3x – 6 + 6 < 9 + 6

-3x < 15; now, divide both sides by -3 and switch sign

-3x/-3 > 15/-3

So x > -5