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 = -21Along 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.
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!
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). After simplification, the variable is isolated. x = (17/3)Back to top.
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.
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).Back to top.
On many occasions, you will be asked to show the solution to an inequality by graphing it on a number line. This is usually covered in elementary algebra (Algebra I) courses. This custom has been followed on this site, so follow this link to understand graphing on a number line.
As with equations, inequalities also have principles dealing with addition and multiplication. They are outlined below.
1. Addition Principle for Inequality — If a > b then a + c > b + c.
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).
1. 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 > -.2One 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.
Absolute value in inequalities is a little more complicated. For example, |x| >= 4 asks us for all numbers that have an absolute value that is greater than or equal to 4. Obviously, 4 and any number greater than 4 is a solution. The confusing part comes from the fact that -4 and any number less than -4 is a solution (|-4| = 4, |-5| = 5, etc.). Therefore, the solution is x >= 4 or x <= -4.
Absolute value becomes even more complicated when dealing with equations. However, there is a theorem that tells us how to deal with equations with absolute value and complicated inequalities.
1. If X is any expression, and b any positive number, and |X| = b it is the same as |X| = b or |X| = -b.
2. If X is any expression, and b any positive number, and |X| < b it is the same as -b < X < b.
3. If X is any expression, and b any positive number, and |X| > b it is the same as X < -b, X > b.
1. 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)Back to top.
Take the quiz on solving equations and inequalities. The quiz is very useful for either review or to see if you've really got the topic down.