Download Example: Solving Inequalities

Representing Inequalities, Solving Inequalities and Absolute Value
Inequalities
Mathematical sentences that use any of the following symbols
 > Greater than
 < Less than
 ≤ Less than or equal to
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 ≥ Greater than or equal to
Solving Inequalities


Done the same way you solve equations.
Exception: when you multiply or divide both sides of an inequality by a negative number, you must
change the direction of the inequality symbol.
Example: Solving Inequalities Using Addition/Subtraction
Solve the following inequalities and graph the solution on the number line.
a. y + 3 > 5
b. x - 3 < 5
Step 1: Isolate y variable
Step 1: Isolate the x variable
Subtract 3 from both sides
add 3 to both sides
y+3>5
x-3<5
- 3 > -3
+ 3 < +3
y >2
x < 9
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
Example: Solving Inequalities Using Multiplication/Division
Solve the following inequalities and graph the solution on the number line.
a. 4y > 12
b. –3y > 15
Step 1: Isolate y variable
divide both sides by 3
4 y 12
>
4
4
y>3
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
Step 1: Isolate the y variable
divide both sides by -3
3 y 15
>
3
3
y < -5 (since we divided by negative, ineq switched)
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
Solving Inequalities (Multi-Step)








Complete the Distributive Property
Simplify by adding like terms.
Eliminate the variable on 1 side
Eliminate constant term on the side with variable
Solve for the variable
Check solution
Remember: addition/subtraction must be done before multiplication/division
Note: some inequalities have no solution and others are true for all real numbers.
Representing Inequalities, Solving Inequalities and Absolute Value
Example: Solving Inequalities (Multi-Step)
Solve the following inequalities and graph the solution on the number line.
a. 2y + 3 < 9
b. 3y + 2y > 15
Step 1: Opposite of add is subtract
So subtract 3 from both sides
Step 1: Add like terms
So add 3y + 2y
Step 2: Perform the necessary operation
2y + 3 < 9
- 3 -3
2y
<6
Step 2: Perform the necessary operation
5y > 15
Step 3: Opposite of multiply is divide
So, divide by 2
Step 3: Opposite of multiply is divide
So, divide by 5
Step 4: Perform the necessary operation
2y 6
<
2
2
y<3
Step 4: Perform the necessary operation
5 y 15
>
5
5
y > 3
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
Compound Inequalities


2 inequalities joined by the word “and” or “or”
Example: -5≤ x ≤ 7 is the same as x ≥ -5 and x ≤ 7
Example: Solving Compound Inequalities (and/or)
Solve the following compound inequalities and graph the solution on the number line.
a. –4 < r –5 ≤ -1
b. 4v + 3 < -5 or –2v + 7 < 1
Step 1: Isolate the variable r
Add 5 to all sides
–4 < r –5 ≤ -1
+5
+5
+5
1<r≤4
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
Step 1: Isolate the variable v
4v + 3 < -5 or –2v + 7 < 1
-3 -3
-7 -7
2v 6
4 v 8
 or
`

4 4
2 2
v < -2 or v > 3
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
Representing Inequalities, Solving Inequalities and Absolute Value
Absolute Value Equations & Inequalities




Since absolute value represents distance, it can never be negative
When solving for |a| = b, 2 solutions a = b and a = -b
When solving for |a| < b, solving for –b < a < b
When solving for |a| > b, solving for a < -b or a > b
Example: Solving Absolute Value Equations
Solve the following equations.
a. | x | + 5 = 11
Step 1: Isolate absolute value function
|x | + 5 – 5 = 11 – 5
Step 2: Simplify
|x| = 6
Step 3: Write 2 equations & solve
x = 6 or x = -6
b. |2p + 5| =11
|2p + 5| = 11
2p + 5 = 11 or 2p + 5 = -11
-5 = -5
-5 = -5
2p = 6
or 2p = -16
p =3
or p = -8
Example: Solving Absolute Value Inequalities
Solve the following inequalities. Check and graph your solution.
a. |n -1 | < 5
b. |v -3| ≥ 4
n – 1 < 5 or n –1 > -5
v – 3 ≥ 4 or v – 3 ≤ -4
n –1 + 1 < 5 + 1 or n –1 + 1 > -5 +1
v – 3 + 3 ≥ 4 + 3 or v –3 + 3 ≤ -4 + 3
n<6
or n > -4
v≥7
or v ≤ -1
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
Representing Inequalities, Solving Inequalities and Absolute Value
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Representing Inequalities Practice
Inequality
1.
5< x < 8
2.
x < -3
3.
x>0
Interval
Graph
Set Notation
Rewrite as an inequality and graph:
4. [5, 9)
5. [-3, 10]
6. (-, 5]
7. (-3, )
Rewrite in interval notation, set notation, and graph:
8. -6 < x < 6
9. x > -3
10. x < -2
11. -2 < x < 5
12. x < 4 or x > 6
13. x < -2 or x > 0
Write as an inequality, in interval notation and in set notation:
14.
15.
-1
-8
3
16.
17.
-4
7
3
18.
18
19.
6
-9
Representing Inequalities, Solving Inequalities and Absolute Value
Practice: Solving Inequalities Using Addition/Subtraction and Multiplication/ Division
Solve the following inequalities. Graph your solution.
1. x – 3 < 5
2. 12 ≤ x – 5
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
3.
n – 7 ≤ -2
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
5.
2
n≤2
3
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
7.
x
< -1
2
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
4.
–4 > b - 1
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
6. 6 ≤
3
w
5
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
8. –20 > -5c
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
Practice: Solving Inequalities (Multi-Step)
Solve the following inequalities and graph the solution on the number line.
9. -15c – 28 > 152
10. 4x – x + 8 ≤ 35
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
Representing Inequalities, Solving Inequalities and Absolute Value
11. 2x – 3 > 2(x-5)
12. 7x + 6 ≤ 7(x – 4)
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
Practice: Solving Compound Inequalities (and/or)
Solve the following compound inequalities and graph the solution on the number line.
13. –6 < 3x < 15
14. –3 < 2x – 1 < 7
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
15. 7 < -3n + 1 ≤ 13
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
17. 2d + 5 ≤ -1 or –2d + 5 ≤ 5
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
16. –2x + 7 > 3 or 3x – 4 ≥ 5
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
18. 3x + 2 < -7 or –4x + 5 < 1
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
Representing Inequalities, Solving Inequalities and Absolute Value
Practice: Solving Absolute Value Equations
Solve the following equations. Check your solution.
19. |t| -2 = -1
20. 3|n| = 15
21. 4 = 3|w| - 2
Practice: Solving Absolute Value Inequalities
Solve the following inequalities. Graph your solution.
22. |w + 2 | > 5
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
24. 4 3s  7  5  7
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
23. |y – 5| ≤ 2
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
25. 2 j  9  2  10
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10