The document provides an algebra review on manipulating algebraic equations to solve for an unknown variable x. It covers topics such as: adding/subtracting numbers, multiplying/dividing numbers, using a combination of operations, dealing with x in the denominator, misconceptions to avoid, exponents, and practice problems. The review explains that any manipulation done to one side of an equation must also be done to the other side to solve for x. It provides examples of isolating the term containing x and then solving.

1. Algebra Review

2. Manipulating Algebraic equations
• Purpose – to solve algebraic equations for an unknown variable
• Key to solving – any manipulation you do to one side of the equation must
be done to the other side of the equation.

3. Numbers added and Subtracted
• Solve the following for x:
x + 5 = 13
• Solution (isolate x):
– To get x as the only item on the left hand side of the equation, we
subtract 5 from both sides of the equation, since it was added to the
side with the x:
x + 5 – 5 = 13 – 5
– Combining like terms we get:
x + 5 – 5 = 13 – 5
x = 8

4. Numbers multiplied and divided
• Solve the following for x:
𝑥
5
= 13
• Solution (isolate x):
– To get x as the only item on the left hand side of the equation, we
multiply both sides of the equation by 5 since the x was divided by 5:
5 ∗ 𝑥
5
= 13 ∗ 5
– Combining like terms we get:
5 ∗ 𝑥
5
= 13 ∗ 5
x = 65

5. Combination of operations
• Solve the following for x:
5x + 6 = 31
• Isolate the term that contains the variable first:
5x + 6 − 6 = 31 − 6
5x = 25
• Now solve for x:
5x
5
=
25
5
x = 5

6. x in the denominator
• When x is in the denominator it is easiest to isolate the x term, then cross multiply
• Solve the following for x:
2
𝑥
−
1
5
= 2.6
• Isolate the term with x first:
2
𝑥
−
1
5
+
1
5
= 2.6 +
1
5
2
𝑥
= 2.6 +
1
5
• Combine like terms:
2
𝑥
= 2.8
• Cross multiply – multiply the denominators by the numerators on the opposite side:
2
𝑥
=
2.8
1
2 ∗ 1 = 2.8 ∗ 𝑥
2 = 2.8𝑥
• Solve for x:
2
2.8
=
2.8𝑥
2.8
0.714 = 𝑥

7. Misconception Alert!
• Be very careful not to accidentally make a variable in the denominator a
numerator.
• For example:
1
𝑥
= 10
– Do not just say x = 10!
– Cross multiply:
1 = 10𝑥
– Solve for x:
1
10
=
10𝑥
10
0.1 = 𝑥

8. Exponents
• When dealing with variables raised to a power, treat them the same and deal with the
exponent at the very end.
– Solve the following for x:
𝑥2
5
= 13
– Solution (isolate x2
, then solve for x):
• To get x as the only item on the left hand side of the equation, we multiply
both sides of the equation by 5 since the x was divided by 5:
5 ∗ 𝑥2
5
= 13 ∗ 5
• Combining like terms we get:
5 ∗ 𝑥2
5
= 13 ∗ 5
𝑥2
= 65
• Take the square root of both sides:
𝑥2 = 65
𝑥 = 8.1

9. Pause and Practice
• Solve each of the following for x:
– 2𝑥 + 5 = 9
–
5
𝑥
= 3
–
1
2𝑥−4
= 9
– 3𝑥2
= 27
–
1
𝑥2 + 4 = 13

10. Pause and Practice
• Solve each of the following for x:
– 2𝑥 + 5 = 9
• 𝑥 = 2
–
5
𝑥
= 3
• 𝑥 = 1.67
–
1
2𝑥−4
= 9
• 𝑥 = 2.06
– 3𝑥2
= 27
• 𝑥 = 3
–
1
𝑥2 + 4 = 13
• 𝑥 = 0.33

11. Try the exercises
• Please check your work before inputting the answers into Blackboard.