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Test PrepACTMathMiscellaneous Algebra

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√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:




Thus, the solution is: no3.PNG OR no4.PNG

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 eq11.PNG


Now just solve for x!



Try the following examples on your own:

Answers and Explanations

  1. 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.
  2. The correct answer is F. First, isolate the absolute value expression: |2x - 3| = 7. Then, split the expression into two equations: 2x - 3 = 7 and 2x - 3 = -7. Solve the first: 2x = 10, so x = 5. Then, solve the second: 2x = -4, so x = -2. Thus, x = -2 or 5. 


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