Module
One:
Solving
Equations and Inequalities
Solving equations is an important concept. In this module we provide step-by-step solutions to solving equations in one variable.
Combining Like Terms
Example 1.
Simplify: 2x + 3 + 4x – 10
Step 1. 2x + 4x + 3 – 10
Step 2. 6x – 7
Example 2.
Simplify: 14a + 8b – 6a – 9b
Step 1. 14a – 6a + 8b – 9b
Step 2. 8a – b
Solving Equations
Example 3.
Solve for x: 2x + 3 + 4x – 10 = 11
Step 1. 2x + 3 + 4x – 10 = 11
Step 2. 6x – 7 = 11
Step 3. 6x – 7 + 7 = 11 + 7
Step 4. 6x = 18
Step 5. 
Step 6. x = 3
To perform a check that 3 is the solution of 2x + 3 + 4x – 10 = 11, substitute 3 for the variable, x, in the original equation and evaluate.
If x = 3, then 2x + 3 + 4x – 10 = 11 becomes
2(3) + 3 + 4(3) – 10 = 11
6 + 3 + 12 – 10 = 11
21 – 10 = 11
11 = 11
Since 11 = 11 is a true statement, 3 is the correct solution.
In the next three examples we multiply both sides of the equation by the lowest common denominator to eliminate the fractions.
Example 4.
Solve for x: 
Step 1. 
Step 2. 3x = – 24
Step 3. x = – 8
Example 5.
Solve for x: 
Step 1. 
Step 2. 7x = 2 – x
Step 3. 8x = 2
Step 4.
Step 5.
Example 6.
Solve for x: 
Step 1. 
Step 2. 4x = 40 + 15x
Step 3. – 11x = 40
Step 4. 
Solving Inequalities
Example 7.
Solve for x: 5x – 1 > 9
Step 1. 5x > 10
Step 2. x > 2
Example 8.
Solve for x: 2x – 5 £ – 7
Step 1. 2x £ – 2
Step 2. x £ – 1
Reminder
When multiplying or dividing both sides of an inequality by a negative number, the inequality symbol must be reversed.
Example 9.
Solve for x: – 8x – 3 > 13
Step 1. – 8x > 16
Step 2. 
Step 2. x < – 2
Note: The inequality symbol is reversed in step 2 only.
Practice Problems.
Solution to problem 1:
x = – 20
2. Solve for x. 8x + 3 – x = 5 + 4x – 2
Solution to problem 2
8x + 3 – x = 5 + 4x – 2
7x + 3 – 4x = 4x + 3 – 4x
3x + 3 = 3
3x + 3 – 3 = 3 – 3
3x = 0
x = 0
3. Solve for x. 3 > 2x – 5
Solution to problem 3:
3 + 5 > 2x – 5 + 5
8 > 2x
4 > x or x < 4
4. Solve for x. – x – 3 £ 4
Solution to problem 4:
– x – 3 + 3 £ 4+ 3
– x £ 7
x ³ – 7
Solution to problem 5:
LCD = 2 · x = 2x
Multiply both sides of the equation by the LCD.
8x = 5x + 6
3x = 6
x = 2
Solution to problem 6:
LCD = 6
Multiply both sides of the equation by the LCD.
3
5y + 9 = 6y
9 = y