# ACT Math Skill Review: Miscellaneous Algebra

There are a few other expressions you should be comfortable with in terms of algebraic manipulation: radical expressions, absolute value and rational expressions.

#### Solving Radical Expressions

The expression 2√*x *is a radical expression because it involves a root; in this case, the square root of *x.* The equation 2√*x* + 7 = 10 is radical because it involves a radical expression. You can solve this equation in the same way you solve other linear equations:

#### Solving Absolute Value Equations and Inequalities

Familiarity with both the concept and notation of absolute value will be helpful in solving math questions. The absolute value of a number is its distance from zero on the number line. The absolute value of the number *x* is denoted |*x|. *For example, |10| = 10 and |-5| = 5. The absolute value of a number is never negative since it represents the value of the number regardless of charge.

You will be expected to work with expressions that involve absolute value. Let’s take a look at an example:

The first thing to do when solving for *x* is to split the inequality into two:

Then isolate *x* for each inequality:

And,

Thus, the solution is: OR

#### Solving Rational Expressions

A rational expression is the quotient of two polynomials (written as a fraction). For example:

It is possible that you will be asked to solve equations or inequalities involving such expressions. Let’s take a look at the following problem:

For what value of *x* is the following equation true?

Your first step should be to put this equation in linear form. You do this by multiplying both sides by

Now just solve for *x*!

#### Examples

Try the following examples on your own:

#### Answers and Explanations

**The correct answer is D.**You’ll need to factor the expression in the numerator and see what you can simplify. Since none of the answer choices are fractions, it’s likely that (*x*- 5) will be one of the factors. Sure enough, it factors to (*x*- 5)(*x*+ 8). Once you cancel what you can, you are left with*x*+ 8. Choice E is a distractor answer.

**The correct answer is F.**First, isolate the absolute value expression: |2*x*- 3| = 7. Then, split the expression into two equations: 2*x*- 3 = 7 and 2*x*- 3 = -7. Solve the first: 2*x*= 10, so*x*= 5. Then, solve the second: 2*x*= -4, so*x*= -2. Thus,*x*= -2 or 5.

- Time Management
- Number Properties
- Fractions
- Arithmetic
- Order of Operations
- Algebraic Manipulation
- Word Problems
- Solving Equations
- Inequalities
- Linear Equations
- Miscellaneous Algebra
- Functions
- Ratios, Proportions and Variations
- Lines & Angles
- Triangles
- Quadrilaterals
- Circles
- Graphs and Coordinate Geometry
- Probability and Statistics
- Trigonometry

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