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Solving InequalitiesAlgebra I

By Ita Rodríguez

Solving Inequalities• An inequality is a mathematical sentence that uses inequality symbols ()

to compare two expressions.• When you use an expression such as at least or at most, you are talking about an

inequality. You can use inequalities to represent situations that involves minimum or maximum amounts.• Equations have a definite solution. Inequalities have infinite number of solutions.• Since it is impossible to list an infinite number of solutions, a number line graph is

used as means of picturing them.

Inequalities and Their GraphsExample:

Inequalities and Their GraphsThe inequalities have two parts to them. The inequality will be true if it satisfies one of its parts, not both.Example:

Inequalities and Their Graphs

• What inequality represents the verbal expression? All real numbers less than or equal to .

Example 1:

𝑥≤−7All real numbers less than or equal to −7


• What inequality represents the verbal expression? 6 less than a number is greater than 13.

Example 2:

𝑘−6¿136 less than a number is greater than 13


• Is the following number a solution of ?

Begin by substituting the value in the inequality. Use order of operations to simplify. Verify if the inequality is true. Not true

Since is not greater than , is not a solution of the inequality.

Example 3: A solution of an inequality is any number that makes the inequality true. For that reason, there are many solutions to an inequality, not just one.


• Is the following number a solution of ?b) Begin by substituting the value in the inequality. Use order of operations to simplify. Verify if the inequality is true. True

Since is greater than , is a solution of the inequality.

Example 4:

Inequalities and Their GraphsA graph can indicate all of the solutions of an inequality.

First draw a number line.Next, ask yourself if can be part of the solution.No, so draw an open circle of the number 1 to indicate it isn’t.Then ask yourself in which direction are the numbers that are less than .Smaller numbers are to the left, so draw a line from the circle to the left.


First draw a number line.Next, ask yourself if can be part of the solution.Yes, so draw a closed circle of the number to indicate it is.Then ask yourself in which direction are the numbers that are greater than.Bigger numbers are to the right, so draw a line from the circle to the right.


Inequality Graph Explanation

The open circle indicates that 1 is not part of the solution. The line indicates that all the numbers to the left are.

The closed circle indicates that is part of the solution along with all the numbers to the right.

The open circle indicates that is not part of the solution. The line indicates that all the numbers to the right are.

The closed circle indicates that is part of the solution along with all the numbers to the left.


What inequality represents the graph?

First ask yourself if the circled number is included with a closed circle.

Yes, the is included so the inequality will either be .

Then ask yourself if the numbers that are included with the line are less than or greater than .

The shaded numbers are greater than .

Now choose any letter for your variable.

The answer is .

Example 5:


What inequality represents the graph?

First ask yourself if the circled number is included with a closed circle.

No, the is not included so the inequality will either be .

Then ask yourself if the numbers that are included with the line are less than or greater than .

The shaded numbers are less than .

Now choose any letter for your variable.

The answer is .

Example 6:

Solving One-Step Inequalities by Adding or Subtracting

S. Graph the solution.

Begin by aSubtracting 10What is the inverse (opposite) of subtracting 10?Adding 10Now add 10 to both sides of the inequality.

Example 7: One-step inequalities are solved the same way one-step equations are solved (by using the inverse operation).



Example 7:



First draw the number line placing the 24 in

the center. Then, follow the steps for graphing.

Example 7:


A club has to sell at least 25 plants for a fund raiser. Club members sell 8 plants on Wednesday and 9 plants on Thursday. What are the possible numbers of plants the club can sell on Friday to meet their goal?

First look for key words to relate the information given.

At least means that 25 is the smallest possible number. Therefore, it means that the numbers that are allowed are 25, 26, 27, 28, 29, …

So at least is represented by the greater than or equal to symbol .

Example 8:

≥258+¿9 𝑓+¿



Now solve the inequality the same way you solved in example 7.

Example 8:



The possible number of plants the club can sell on Friday to meet their goals are greater than or equal to 8.

Example 8:

Solving One-Step Inequalities by Multiplying or Dividing

What are the solutions to ? Graph the solution.

Begin by aDividing 3What is the inverse (opposite) of dividing 3?Multiplying 3Now multiply 3 to both sides of the inequality.

Example 9:



Example 9:


You walk dogs in your neighborhood after school. You earn $4.50 per dog. How many dogs do you need to walk to earn at least $75? Round to whole numbers.

First look for key words to relate the information given.

$4.50 per dog means multiply $4.50 to every dog you walk.

At least means that $75 is the smallest possible number. Therefore, it means that the numbers that are allowed are 75, 76, 77, 78, 79, …

So at least is represented by the greater than or equal to symbol .

Example 10:

≥75𝑑4.50



Now solve the inequality.

Example 10:



You need to walk at least 17 dogs to earn at least $75. You cannot walk at least 16.67 dog because you cannot have part of a dog.

Example 10:

Solving One-Step Inequalities by Multiplying or DividingYou have already seen solving inequalities using multiplication or division of a positive number.Solving inequalities using multiplication or division of a negative number is different. When you multiply 2 to both sides, the inequality remains true.

TrueHowever, if you multiply to both sides, the inequality becomes false.

False

Solving One-Step Inequalities by Multiplying or DividingTherefor, the rule of solving inequalities when multiplying or dividing by a negative number is to reverse (or flip) the direction of the inequality symbol.



Begin by aMultiplying What is the inverse (opposite) of multiplying ?Dividing which really means to multiply the reciprocal Now multiply to both sides of the inequality.

Example 11:



Flip the symbol

Example 11:



Example 11:

Solving Multi-Step InequalitiesYou solve multi-step inequalities the same way you solve multi-step equations. Use the inverse of the order of operations.

Remember to reverse (flip) the symbol if you multiply or divide by a negative number.

Solving Multi-Step Inequalities

What are the solutions to ?

Begin by looking for an addition or subtraction.5 is being added so the inverse is to subtract 5.

Example 12:



Next, look for a multiplication or division. is being multiplied so the inverse is to divide Since the number you are dividing is negative, you will have to flip the symbol.

Example 12:



You can rewrite the inequality with the variable first. Notice how the symbol points to the variable, so when you rewrite it, make sure the symbol still points to the variable.

Example 12:


What are the solutions to )?

Begin by distributing the 2. )Next, move the variable to the left side. Since the variables cancel out and the inequality remains true, the solution is all real numbers.

Example 13:



Begin by combining like terms. Next, move the variable to the left side. Since the variables cancel out and the inequality is false, there is no solution to the inequality.

Example 14:



Begin by distributing the 2. )Next, move the variable to the left side.

Example 15:



Then, move the constant to the right.

Example 15:



is being multiplied so the inverse is to divide Since the number you are dividing is negative, you will have to flip the symbol.

Example 15:

Solving Compound InequalitiesA compound inequality consists of two distinct inequalities joined by the word or the word .The graph of compound inequality with the word contains the overlapping region of two inequalities.

Solving Compound InequalitiesThere is another way to write an compound inequality. Guide yourself with the graph.

is the same as .

3¿𝑥 7≤

Solving Compound InequalitiesThe graph of compound inequality with the word contains each graph of the two inequalities.

or

Solving Compound Inequalities

What are the solutions to or ? Graph the solution.

Begin by solving each inequality separately.

Example 16:



Now continue with the second inequality.

Example 16:



Example 16:


What are the solutions to or ? Graph the solution. or

Example 16:



You can solve this compound inequality two ways.Method one, separate it into two inequalities and work them out as in example 16.Method two, work out its two parts at the same time.

Example 17:



Method one:Separate it into two inequalities.

Continue to solve normally.

Example 17:

8≥−5 𝑥−2 −5 𝑥−2>3



Now continue with the second inequality.

Example 17:



Example 17:



Now put the two inequalities back together following the order of the number line and graph.

3

Example 17:



Method two:Work out its two parts at the same time. Focus your attention in the center where thevariable is. You are subtracting a 2 so the inverse is to add 2. Do it to the three parts.

Example 17:

10+2 +2+2

10≥−5 𝑥>55



Method two:Again, focus your attention in the center where thevariable is. You are multiplying so the inverse is to divide . Do it to the three parts.Remember to flip the symbols because you aredividing by a negative number.Graph as we did before.

Example 17:

10≥−5 𝑥>510−5 ≥

−5 𝑥−5 >

5−5

−2≤ 𝑥<−1

The End