Solving inequalities

Solving Inequalities
Algebra 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 Graphs
Example:
• 3 < 5 3 𝑖𝑠 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 5
• 0 > −2 0 𝑖𝑠 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 − 2
• 5 ≤ 6 5 𝑖𝑠 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 𝑜𝑟 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 6
• 5 ≤ 5 5 𝑖𝑠 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 𝑜𝑟 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 5
• 7 ≥ −1 7 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 𝑜𝑟 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 − 1
• 7 ≥ 7 7 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 𝑜𝑟 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 7

The inequalities ≤ 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 𝑜𝑟 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝑎𝑛𝑑 ≥ 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 𝑜𝑟 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 have
two parts to them. The inequality will be true if it satisfies one of its parts, not
both.
Example:
• 5 ≤ 6 𝑇ℎ𝑖𝑠 𝑖𝑛𝑒𝑞𝑢𝑎𝑙𝑖𝑡𝑦 𝑖𝑠 𝑡𝑟𝑢𝑒 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 5 𝑖𝑠 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 6.
• 5 ≤ 5 𝑇ℎ𝑖𝑠 𝑖𝑛𝑒𝑞𝑢𝑎𝑙𝑖𝑡𝑦 𝑖𝑠 𝑡𝑟𝑢𝑒 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 5 𝑖𝑠 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 5.
• 7 ≥ −1 𝑇ℎ𝑖𝑠 𝑖𝑛𝑒𝑞𝑢𝑎𝑙𝑖𝑡𝑦 𝑖𝑠 𝑡𝑟𝑢𝑒 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 7 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 − 1.
• 7 ≥ 7 𝑇ℎ𝑖𝑠 𝑖𝑛𝑒𝑞𝑢𝑎𝑙𝑖𝑡𝑦 𝑖𝑠 𝑡𝑟𝑢𝑒 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 7 𝑖𝑠 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 7.

• What inequality represents the verbal expression?
All real numbers 𝑥 less than or equal to −7.
Example 1:
𝑥 ≤ −7
All real numbers 𝑥 less than or equal to −7

• What inequality represents the verbal expression?
6 less than a number 𝑘 is greater than 13.
Example 2:
𝑘 − 6 > 13
6 less than a number 𝑘 is greater than 13

• Is the following number a solution of 2𝑥 + 1 > −3?
a) −3
Begin by substituting the value in the inequality. 2 −3 + 1 > −3
Use order of operations to simplify. −6 + 1 > −3
Verify if the inequality is true. −5 > −3 Not true
Since −5 is not greater than −3, −3 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 2𝑥 + 1 > −3?
b) 1
Begin by substituting the value in the inequality. 2 1 + 1 > −3
Use order of operations to simplify. 2 + 1 > −3
Verify if the inequality is true. 3 > −3 True
Since 3 is greater than −3, 1 is a solution of the inequality.
Example 4:

A graph can indicate all of the solutions of an inequality.
𝑛 < 1 (𝑛 𝑖𝑠 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 1)
−3 − 2 − 1 0 1 2 3
First draw a number line.
Next, ask yourself if 1 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 1.
Smaller numbers are to the left, so draw a line from the circle to the left.

𝑎 ≥ 𝑜 (𝑎 𝑖𝑠 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 𝑜𝑟 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 0)
−3 − 2 − 1 0 1 2 3
First draw a number line.
Next, ask yourself if 0 can be part of the solution.
Yes, so draw a closed circle of the number 0 to indicate it is.
Then ask yourself in which direction are the numbers that are greater than 0.
Bigger numbers are to the right, so draw a line from the circle to the right.

Inequality Graph Explanation
𝑛 < 1
−3 − 2 − 1 0 1 2 3
The open circle indicates that 1 is not
part of the solution. The line indicates
that all the numbers to the left are.
𝑎 ≥ 𝑜
−3 − 2 − 1 0 1 2 3
The closed circle indicates that 0 is part
of the solution along with all the
numbers to the right.
𝑓 > −3
−3 − 2 − 1 0 1 2 3
The open circle indicates that −3 is not
part of the solution. The line indicates
that all the numbers to the right are.
−2 ≥ 𝑥
−3 − 2 − 1 0 1 2 3
The closed circle indicates that −2 is
part of the solution along with all the
numbers to the left.

What inequality represents the graph?
−3 − 2 − 1 0 1 2 3
First ask yourself if the circled number is included with a closed circle.
Yes, the −2 is included so the inequality will either be ≤ 𝑜𝑟 ≥.
Then ask yourself if the numbers that are included with the line are less than −2 or greater than −2 .
The shaded numbers are greater than −2 .
Now choose any letter for your variable.
The answer is 𝑓 ≥ −2.
Example 5:

What inequality represents the graph?
−3 − 2 − 1 0 1 2 3
First ask yourself if the circled number is included with a closed circle.
No, the 3 is not included so the inequality will either be < 𝑜𝑟 >.
Then ask yourself if the numbers that are included with the line are less than 3 or greater than 3.
The shaded numbers are less than 3.
Now choose any letter for your variable.
The answer is 𝑚 < 3.
Example 6:

Solving One-Step Inequalities by Adding or
Subtracting
Solve 𝑛 − 10 > 14. Graph the solution.
Begin by asking yourself what is being done to the variable.
Subtracting 10
What is the inverse (opposite) of subtracting 10?
Adding 10
Now 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).

Subtracting
𝑛 − 10 > 14
+10 + 10
24
𝑛 > 24
Example 7:

Subtracting
𝑛 > 24
First draw the number line placing the 24 in
the center. Then, follow the steps for graphing.
23 24 25
Example 7:

Subtracting
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+

Subtracting
Now solve the inequality the same way you solved in example 7.
8 + 9 + 𝑓 ≥ 25
17 + 𝑓 ≥ 25
−17 − 17
𝑓 ≥ 8
Example 8:

Subtracting
𝑓 ≥ 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
𝑥
3
≤ −2? Graph the solution.
Dividing 3
What is the inverse (opposite) of dividing 3?
Multiplying 3
Now multiply 3 to both sides of the inequality.
Example 9:

or Dividing
𝑥
3
𝑥
3
≤ −2
3 ∙
𝑥
3
≤ −2 ∙ 3
𝑥 ≤ −6
Example 9:

or Dividing
𝑥
3
𝑥 ≤ −6
−7 − 6 − 5
Example 9:

or Dividing
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

or Dividing
Now solve the inequality.
4.50𝑑 ≥ 75
4.50𝑑
4.50
≥
75
4.50
𝑑 ≥ 16.67
𝑑 ≥ 17
Example 10:

or Dividing
𝑑 ≥ 17
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:

or Dividing
You 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.
3 < 4
When you multiply 2 to both sides, the inequality remains true.
2 ∙ 3 < 2 ∙ 4
6 < 8 True
However, if you multiply −2 to both sides, the inequality becomes false.
−2 ∙ 3 < −2 ∙ 4
−6 < −8 False

or Dividing
Therefor, the rule of solving inequalities when multiplying or dividing by a negative
number is to reverse (or flip) the direction of the inequality symbol.
3 < 4
−2 ∙ 3 < −2 ∙ 4
−6 > −8 True

or Dividing
What are the solutions to −
3
4
𝑤 > 3? Graph the solution.
Multiplying −
3
4
What is the inverse (opposite) of multiplying −
3
4
?
Dividing −
3
4
which really means to multiply the reciprocal −
4
3
Now multiply −
4
3
to both sides of the inequality.
Example 11:

or Dividing
3
4
−
3
4
𝑤 > 3
−
4
3
∙ −
3
4
𝑤 > 3 ∙ −
4
3
Flip the symbol
𝑤 < −
12
3
Example 11:

or Dividing
3
4
𝑥 < −
12
3
𝑤 < −4
−5 − 4 − 3
Example 11:

Solving Multi-Step Inequalities
You 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.

What are the solutions to −4 ≤ 5 − 3𝑛?
−4 ≤ 5 − 3𝑛
Begin by looking for an addition or subtraction.
5 is being added so the inverse is to subtract 5.
−4 ≤ 5 − 3𝑛
−5 − 5
−9
−9 ≤ −3𝑛
Example 12:

−9 ≤ −3𝑛
Next, look for a multiplication or division.
−3 is being multiplied so the inverse is to divide −3. Since the number you are
dividing is negative, you will have to flip the symbol.
−9
−3
≤
−3𝑛
−3
3 ≥ 𝑛
Example 12:

3 ≥ 𝑛
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.
𝑛 ≤ 3
Example 12:

What are the solutions to 10 − 8𝑎 ≥ 2(5 − 4𝑎)?
Begin by distributing the 2. 10 − 8𝑎 ≥ 2(5 − 4𝑎)
Next, move the variable to the left side. 10 − 8𝑎 ≥ 10 − 8𝑎
+8𝑎 + 8𝑎
10 ≥ 10
Since the variables cancel out and the inequality remains true, the solution is all
real numbers.
Example 13:

What are the solutions to 6𝑚 − 5 > 7𝑚 + 7 − 𝑚?
Begin by combining like terms. 6𝑚 − 5 > 7𝑚 + 7 − 𝑚
Next, move the variable to the left side. 6𝑚 − 5 > 6𝑚 + 7
−6m − 6𝑚
−5 > 7
Since the variables cancel out and the inequality is false, there is no solution to the
inequality.
Example 14:

What are the solutions to 5 − 3𝑓 > 2(4𝑓 − 3)?
Begin by distributing the 2. 5 − 3𝑓 > 2(4𝑓 − 3)
Next, move the variable to the left side. 5 − 3𝑓 > 8𝑓 − 6
−8𝑓 − 8𝑓
5 − 11𝑓 > −6
Example 15:

Then, move the constant to the right. 5 − 11𝑓 > −6
−5 − 5
−11𝑓 > −11
Example 15:

−11𝑓 > −11
−11 is being multiplied so the inverse is to divide −11. Since the number you are
dividing is negative, you will have to flip the symbol.
−11𝑓
−11
>
−11
−11
𝑓 < 1
Example 15:

Solving Compound Inequalities
A compound inequality consists of two distinct inequalities joined by the word 𝑎𝑛𝑑
or the word 𝑜𝑟.
The graph of compound inequality with the word 𝑎𝑛𝑑 contains the overlapping
region of two inequalities.
𝑥 > 3
1 2 3 4 5 6 7
𝑥 ≤ 7
1 2 3 4 5 6 7
𝑥 > 3 𝑎𝑛𝑑 𝑥 ≤ 7
1 2 3 4 5 6 7

There is another way to write an 𝑎𝑛𝑑 compound inequality. Guide yourself with
the graph.
𝑥 > 3 𝑎𝑛𝑑 𝑥 ≤ 7
1 2 3 4 5 6 7
𝑥 > 3 𝑎𝑛𝑑 𝑥 ≤ 7 is the same as 3 < 𝑥 ≤ 7.
3 < 𝑥 7≤

The graph of compound inequality with the word 𝑜𝑟 contains each graph of the
two inequalities.
𝑥 ≤ −1
−3 − 2 − 1 0 1 2 3
𝑥 > 2
−3 − 2 − 1 0 1 2 3
𝑥 ≤ −1 or 𝑥 > 2
−3 − 2 − 1 0 1 2 3

What are the solutions to 3𝑡 + 2 < −7 or −4𝑡 + 5 < 1? Graph the solution.
Begin by solving each inequality separately.
3𝑡 + 2 < −7 −4𝑡 + 5 < 1
−2 − 2
−9
3𝑡 < −9
3
3
𝑡 <
−9
3
Example 16:

3
3
𝑡 <
−9
3
− 4𝑡 + 5 < 1
𝑡 < −3
Now continue with the second inequality.
−4𝑡 + 5 < 1
−5 − 5
−4
Example 16:

𝑡 < −3 − 4𝑡 + 5 < 1
−5 − 5
−4
−4𝑡 < −4
−4
−4
𝑡 <
−4
−4
𝑡 > 1
Example 16:

𝑡 < −3 or 𝑡 > 1
−4 − 3 − 2 − 1 0 1 2
Example 16:

What are the solutions to 8 ≥ −5𝑥 − 2 > 3? Graph the solution.
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. 8 ≥ −5𝑥 − 2 > 3
Continue to solve normally.
+2 + 2
10
10 ≥ −5𝑥
Example 17:
8 ≥ −5𝑥 − 2 −5𝑥 − 2 > 3

10 ≥ −5𝑥 −5𝑥 − 2 > 3
10
−5
≥
−5𝑥
−5
−2 ≤ 𝑥
Now continue with the second inequality.
Example 17:

−2 ≤ 𝑥 −5𝑥 − 2 > 3
+2 + 2
5
−5𝑥 > 5
−5𝑥
−5
>
5
−5
𝑥 < −1
Example 17:

−2 ≤ 𝑥 𝑥 < −1
Now put the two inequalities back together following the order of the number line
and graph.
−2 ≤ 𝑥 < −1
−3 − 2 − 1 0 1 2 3
Example 17:

Method two:
Work out its two parts at the same time. 8 ≥ −5𝑥 − 2 > 3
Focus your attention in the center where the
variable 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 ≥ −5𝑥 > 5
5

Method two:
Again, focus your attention in the center where the
variable is. You are multiplying −5 so the
inverse is to divide −5. Do it to the three parts.
Remember to flip the symbols because you are
dividing by a negative number.
Graph as we did before.
Example 17:
10 ≥ −5𝑥 > 5
10
−5
≥
−5𝑥
−5
>
5
−5
−2 ≤ 𝑥 < −1

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