Solving Equations & Inequalities
Solving Equations



In solving equations, 1 must view an equation as a balanced scale. What you do to one side, you must
do to the other side.
Think of an equation as untying a knot. In order to undo the knot, you must reverse the order in which
you tied the knot. Hence, inverse operation.
Strategies for solving equation
o Simplify by adding like terms.
o Solve for the variable
o Eliminate the variable on 1 side
o Check solution
o Eliminate constant term on the side with variable
Definitions:

Equivalent Equations: equations that have the same solution.

Inverse Operation: Operation that undoes another operation. Example addition/subtraction,
multiplication/division

Reciprocal/Multiplicative Inverse: Flipping/switching the denominator and the numerator.
Key Concepts

The product of any number and its reciprocal is 1.
Example: Solve Equation (w/ Fractions)
Remember the following hints when dealing with fractions in an equation:
o Convert any mixed numbers into improper fractions.
 Multiply the whole number with the denominator and add the numerator. That becomes
the new numerator.
 Put the new numerator over the denominator.
1
3 * 2 1 7
 Example: 3 becomes
=
2
2
2
o If the equation has a fraction times a variable, multiply both sides of equation by the
reciprocal.
2
2
3
 Example: y = 12 (reciprocal of
is )
3
3
2
3 2
3
* y = 12 *
2 3
2
36
y=
= 18
2
o When add/subtracting fractions, make sure to have common denominators.
3
7
 Example: n +
=
4
8
3
3
=4
4
3
(Need to change
to fraction with denominator of 8. In order to do this, simply multiply both
4
numerator & denominator by 2)
1
Rev A
Solving Equations & Inequalities
7 6
1
=
8 8
8
1
3
1
6
7
Check: +
= + =
8
4
8
8
8
n
=
Practice: Solve Equation (w/ Fractions)
1.
1
1
a + 1 = 10
2
2
2.
2
x + 4 = 18
3
3.
x
+3=8
5
Example: Solve Equation (w/ variables on both sides)
Remember the following when dealing with variables on both sides:
 Simplify by adding like terms.
 Eliminate the variable on 1 side
 Eliminate constant term on the side with variable
 Solve for the variable
1. 6x + 3 = 8x – 21
2. 4p –10 = p + 3p – 2p
Step 1: Move 6x to other side.
6x + 3 = 8x – 21
-6x
-6x
3 = 2x – 21
Step 1: Combine like terms
4p – 10 = 2p
Step 2: Move –21 to other side
3 = 2x – 21
+21
+ 21
24 = 2x
Step 2: Move 4p to other side
4p – 10 = 2p
-4p
- 4p
-10 = -2p
Step 3: Divide by 2 on both sides
24 2 x
=
2
2
Step 3: Divide by –2 on both sides
10 2 p
=
2
2
12 = x
5=p
Step 4: Check solution
6(12) + 3 = 8(12) – 21
72 + 3 = 96 – 21
75 = 75
Step 4: Check solution
4(5) – 10 = 5 + 3(5) – 2(5)
20 – 10 = 5 + 15 - 10
10 = 10
Practice: Solve Equation (w/ variables on both sides)
4. 6x - 2 = x + 13
5. 5y – 3 = 2y + 12
2
6. 4k – 3 = 3k + 4
Rev A
Solving Equations & Inequalities
Solving Formulas for indicated variables


Follow same steps as solving an equation.
If there are any variables in the denominator, then restrictions must be stated. Remember, you cannot
divide by 0.
Examples: Solving Formulas for indicated variables
Solve each equation for x. Find any restrictions.
1. ax + bx -15 = 0
2. d =
2x
+b
a
Inequalities
Mathematical sentences that use any of the following symbols
 > Greater than
 < Less than
 ≤ Less than or equal to
 ≥ Greater than or equal to
Example: Identifying Solution by Evaluating
Is each number a solution of 2 –5x > 13?
a. 3
Step 1: Substitute for x 2 – 5(3) > 13
b. -4
2 – 5(-4) > 13
Step 2: Simplify
2 – 15 > 13
2 + 20 > 13
Step 3: Compare
– 13 > 13
3 doesn’t make original inequality true
22 > 13
-4 makes the original inequality true
Practice: Identifying Solution by Evaluating
Is each number a solution of 6x – 3 > 10?
1. 1
2. 3
no
yes
Graphing Inequalities on Number Line


If = symbol included, circle is closed. Circle open in all other cases.
Arrow is pointed either right or left, depending on inequality symbol used.
Example: Graphing Inequalities
Practice: Graphing Inequalities
Graph the following inequalities on the number line.
c. x < 3
Graph the following inequalities on the number line.
3. m ≥ 4
–5
–4
–3
–2
–1
0
1
2
3
4
–5
5
–4
–3
–2
–1
0
1
2
3
4
5
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.
3
Rev A
Solving Equations & Inequalities
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
Step 2: Check with number > 2
3+3>5
6>5
Step 2: Check with a number < 8
5–3<5
2<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
Practice: Solving Inequalities Using Addition/Subtraction
Solve the following inequalities. Graph and check your solution.
4. x – 3 < 5
5. 12 ≤ x – 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
6. n – 7 ≤ -2
7. –4 > b - 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
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
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)
Step 2: Check with number > 3
4(5) > 12
20 > 12
Step 2: Check with a number < -5
-3(-6) > 15
18 > 15
–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
Rev A
Solving Equations & Inequalities
Practice: Solving Inequalities Using Multiplication/Division
Solve the following inequalities. Graph and check your solution.
2
3
8.
n≤2
9. 6 ≤
w
3
5
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
10.
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
x
< -1
2
11. –20 > -5c
–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
Solving Inequalities (Multi-Step)







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.
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
5
Rev A
Solving Equations & Inequalities
Step 5: Check Solution
2(1) + 3 < 9
2+3<9
5<9
Step 5: Check Solution
3(4) + 2(4) > 15
12 + 8 > 15
20 > 15
–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 Inequalities (Multi-Step)
Solve the following inequalities and graph the solution on the number line.
12. -15c – 28 > 152
13. 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
14. 2x – 3 > 2(x-5)
15. 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
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
Step 1: Isolate the variable v
Add 5 to all sides
4v + 3 < -5 or –2v + 7 < 1
–4 < r –5 ≤ -1
-3 -3
-7 -7
4 v 8
2 v 6
 or

+5 +5 +5
`
4 4
2  2
1<r≤4
v < -2 or v > 3
Step 2: check solution between 1 and 4
-4 < 2 –5 ≤ -1
-4 < -3 ≤ -1
Checks out
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
Step 2: check solutions
4(-3) + 3 < -5
-2(4) +7 < 1
-12 + 3 < -5
-8 + 7 < 1
-9 < -5
-1 < 1
Checks out
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
6
Rev A
Solving Equations & Inequalities
Practice: Solving Compound Inequalities (and/or)
Solve the following compound inequalities and graph the solution on the number line.
16. –6 < 3x < 15
17. –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
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
18. 7 < -3n + 1 ≤ 13
19. –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
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
20. 2d + 5 ≤ -1 or –2d + 5 ≤ 5
21. 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
–10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
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. Check your solution.
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
Step 4: Check Solutions
|6| + 5 = 11 and |-6| + 5 = 11
6 + 5 = 11
6 + 5 = 11
7
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
|2(3) + 5| = 11 and |2(-8) + 5| = 11
|6 + 5| = 11 and |-16 + 5| = 11
Rev A
Solving Equations & Inequalities
Practice: Solving Absolute Value Equations
Solve the following equations. Check your solution.
1. |t| -2 = -1
2. 3|n| = 15
3. 4 = 3|w| - 2
Example: Solving Absolute Value Inequalities
Solve the following inequalities. Check and graph your solution.
c. |n -1 | < 5
d. |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
Practice: Solving Absolute Value Inequalities
Solve the following inequalities. Check and graph your solution.
4. |w + 2 | > 5
5. |y – 5| ≤ 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
Examples: Real-world Applications
1. The marching band agrees to play for $200 plus 25% of the ticket sales. Find the ticket sales needs for the band
to receive at least $500.
Step 1: Define variables:
X = ticket sales
Step 2: Write equation:
200 + 0.25x ≥ 500
Step 3: Solve equation
0.25x ≥ 300; x ≥ 1200
Step 4: Check solution
200 + 0.25(1200) ≥ $500
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 a monthly income of at least $1800?
8
Rev A