SECTION 1-4: Solving Inequalities

We solve inequalities the same way we solve equations with the following

exception:

**GOLDEN RULE for Inequalities**

When you multiply or divide by a __________ number, you MUST _______ the direction of the inequality symbol.

The Inequality Symbols

Key words that describe each symbol:1. < - less than,

2. > - greater than,

3. ≤ - less than or equal to,

4. ≥ - greater than or equal to,

Solving Inequalities - EXAMPLES

EX.A) -3x > 6 B) 2 - x ≤ -79

1

Graphing Solutions of Inequalities Rules for graphing inequalities:

< or > - use an ________ dot≤ or ≥ - use a _________ dot

< or ≤ - shade to the ________> or ≥ - shade to the ________

** The variable must be on the _______ after you solve to use these rules!! (Ex. x < 3)

Graph the solution: EXAMPLES

EX.A) 3x – 12 < 3

Graph: ------------------------------------

Is ___ part of the solution?

Check your answer

How can we check our answer to EX.A if 5 is not part of the solution??

EXAMPLES – Graphing the Solution

EX.B) 9 – 2x > 5 EX.C) 3x – 7 ≤ 5

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ALL REAL NUMBERS & NO SOLUTION

When our result has no variable left in it, our answer is either all real numbers or no solution.

If the result is _______ (Ex. 3 < 7), our answer is ________________________________.

If the result is _______ (Ex. 3 > 7), our answer is ________________________________.

EXAMPLES

EX.1) 2x – 3 > 2(x – 5)

Our result is ______. Therefore, our answer is ___________________________.

Graph: ----------------------------------

EXAMPLES

EX.2) 7x + 6 < 7(x – 4)

Our result is ______, therefore our answer is _______________________.

Graph: --------------------------------

EXAMPLES – Try These:

1) 2x < 2(x + 1) + 3

2) 4(x – 3) + 7 ≥ 4x + 1

3) 4x + 8 > -4(x – 8)

INEQUALITY WORD PROBLEMS - write an inequality for the situation

EX. A band agrees to play for $200 plus 25% of the ticket sales. Find the ticket sales needed for the band to receive at least $500.

Define variables: Let x = __________________

In words, $200 + 25% ticket sales _______ $500

Write an inequality:

Inequality word problems…

Solve the inequality:

Write a sentence for your answer: ______________________________________________________________________________________________________________________________

Inequality word problems…Example 2

A salesperson earns a salary of $700 per month plus 2% of the sales. What must the sales be if the salesperson is to have monthly income of at least $1800.

Let x = _____________________________

Write an equation:

Example 2, continued…

Solve the inequality:

Write a sentence for your answer: _________

_____________________________________________________________________________________________________________________

Example 3

The lengths of the sides of a triangle are 3:4:5. What is the length of the longest side if the perimeter is not more than 84 cm?

Use x to represent the ratio. s1 =

s2 =

s3 =

Example 3, continued…

Write an inequality from the given information:

What is the length of the

longest side??

COMPOUND INEQUALITIES

Compound inequalities are ________ of inequalities joined by _______ or ________.

If ‘and’ and ‘or’ are not written, use the following rule:

Less thAN (<, ≤) use ANd

GreatOR (>, ≥) use OR

‘AND’ Graphs

AND represents the overlap, also called the ___________ of the two inequalities.

We need to transfer everything with 2 lines above onto our final graph.

EX. -----------------------------------

EX. -----------------------------------

‘AND’ Examples

3x – 1 > -28 AND 2x + 7 < 19STEP 1: Solve each inequality separately

Step 2: Graph each above the final number line

Step 3: ----------------------------------

‘AND’ Examples

2x < x + 6 < 4x – 18 (less thAN use AND)

STEP 1: Solve each inequality separately

Step 2: Graph each above the final number line

Step 3: ----------------------------------

‘OR’ Graphs

OR represents the ________ of the two inequalities.

We need to transfer everything with 1 or more lines above onto our final graph.

EX. -----------------------------------

EX. -----------------------------------

‘OR’ Examples

4y – 2 ≥ 14 OR 3y – 4 ≤ -13STEP 1: Solve each inequality separately

Step 2: Graph each above the final number line

Step 3: ----------------------------------

‘OR’ Examples

x - 12 ≥ -5x ≥ -2x – 9 (greatOR use OR)STEP 1: Solve each inequality separately

Step 2: Graph each above the final number line

Step 3: ----------------------------------