On this page, we hope to clear up problems that you might have
with solving equations and/or inequalities. Equations are
the most common thing you'll see in all of math! Click
any of the links below or start scrolling down to begin understanding
equations and inequalities.
Dealing with fractions or decimals
Quiz on Solving Equations and Inequalities
To solve an equation, you isolate the variable are solving for. The Addition Principle says that when a = b, a + c = b + c for any number c. Example:
1. Solve: x + 6 = -15 Solution: Using the Addition Principle, add -6 to each side of the equation. x + 6 - 6 = -15 - 6 The variable is now isolated. x = -21
Along the same lines, the Multiplication Principle says that if a = b and c is any number, a * c = b * c. This principle is also used to help isolate the variable you are asked to solve for. Example:
2. Solve: 4x = 9 Solution: Using the Multiplication Principle, multiply each side of the equation by (1/4). (1/4)4x = (1/4)9 The variable is now isolated. x = (9/4)
Also, be aware of problems where you might need to use both of these principles together! Example:
3. Solve: 3x - 4 = 13 Solution: Use the Addition Principle to add 4 to each side. 3x - 4 + 4 = 13 + 4 After simplifying, 3x = 17. Use the Multiplication Principle to multiply each side by (1/3). (1/3)3x = (1/3)17 After simplification, the variable is isolated. x = (17/3)
When an equation contains fractions or decimals, it usually makes it easier to solve them when the fractions or decimals aren't there. The Multiplication Principle is used to do this. Example:
1. Solve: (3/4)x + (1/2) = (3/2) Solution: Multiply both sides by the LCM of the denominators, 4 in this case. Use the Distributive Law, which says a(b + c) = ab + ac to make the equation easier to deal with. (4 * (3/4)x) + (4 * (1/2)) = 4(3/2) After simplification, you get an equation with no fractions! 3x + 2 = 6 (It's left up to you to solve for x.)
If a and b are real numbers, and ab = 0, either a, b, or both equal 0. This principle, called The Principle of Zero Products, is useful when you have an equation to solve that has two instances of a variable, such as (x + 3)(x - 2) = 0. Example:
2. Solve: 7x(4x + 2) = 0 Solution: Using the Principle of Zero Products, 7x = 0 and 4x + 2 = 0 Solve each equation for x. x = 0 and x = -(1/2) The solutions are 0 and -(1/2).
Math problems containing <, >, <=, and >= are called
inequalities. A solution to any inequality is any number
that makes the inequality true.
1. Solve: x + 3 > 6 Solution: Using the Addition Principle, add -3 to each side of the inequality. x + 3 - 3 > 6 - 3 After simplification, x > 3.
2. Multiplication Principle for Inequalities - If a >b and c is positive, then ac > bc. If a > b and c is negative, then ac < bc (notice the sign was reversed). Example:
2. Solve: -4x < .8 Solution: Using the Multiplication Principle, multiply both sides of the inequality by -.25. Then reverse the signs. -.25(-4x) > -.25(.8) x > -.2
One thing in math that seems to give people trouble throughout
their math careers is absolute value. The absolute
value of any number is its numerical value (ignore the sign).
For example, the absolute value of -6 is 6 and |+3| (the
vertical lines stand for absolute value) is 3.
3. Solve: |5x - 4| = 11 Solution: Use the theorem stated above to rewrite the equation. |X| = b X = 5x - 4 and b = 11 5x - 4 = 11, 5x - 4 = -11 Solve each equation using the Addition Principle and the Multiplication Principle. 5x = 15, 5x = -7 x = 3, x = -(7/5)
Take the Quiz on solving equations and inequalities. (Very useful to review or to see if you've really got this topic down.) Do it!