Algebra II For Dummies. Sterling Mary Jane
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problem with fractions, like cats, is that they aren’t particularly easy to deal with. They always insist on having their own way – in the form of common denominators before you can add or subtract. (Or, with cats, they get hissy.) And division? Don’t get me started!
To solve
Now you distribute the reduced numbers over each parenthesis, combine the like terms, and solve for x:
When you see only one variable in an equation, you have a pretty clear idea what you’re solving for. When you have an equation like 4x + 2 = 11 or 5(3z – 11) + 4z = 15(8 + z), you identify the one variable and start solving for it.
Life isn’t always as easy as one-variable equations, however. Being able to solve an equation for some variable when it contains more than one unknown can be helpful in many situations. If you’re repeating a task over and over – such as trying different widths of gardens or diameters of pools to find the best size – you can solve for one of the variables in the equation in terms of the others.
The equation
If you want to construct a trapezoid that has a set area, you need to figure out what dimensions give you that area. You’ll find it easier to do the many computations if you solve for one of the components of the formula first – for h, b1, or b2.
To solve for h in terms of the rest of the unknowns or letters, you multiply each side by two, which clears out the fraction, and then divide by the entire expression in the parentheses:
You can also solve for b2, the measure of the second base of the trapezoid. To do so, you multiply each side of the equation by two, and then divide each side by h.
Next, subtract b1 from each side of the equation.
Paying off your mortgage with algebra
A few years ago, one of my mathematically challenged friends asked me if I could help her figure out what would happen to her house payments if she paid $100 more each month on her mortgage. She knew that she’d pay off her house faster, and she’d pay less in interest. But how long would it take and how much would she save? I created a spreadsheet and used the formula for an amortized loan (mortgage). I made different columns showing the principal balance that remained (solved for P) and the amount of the payment going toward interest (solved for the difference), and I extended the spreadsheet down for the number of months of the loan. We put the different payment amounts into the original formula to see how they changed the total number of payments and the total amount paid. She was amazed. I was even amazed! She’s paying off her mortgage much sooner than expected!
Linear Inequalities: Algebraic Relationship Therapy
Equations – statements with equal signs – are one type of relationship or comparison between things; they say that terms, expressions, or other entities are exactly the same. An inequality is a bit less precise. Algebraic inequalities show relationships between two numbers, a number and an expression, or between two expressions. In other words, you use inequalities for comparisons.
Inequalities in algebra are less than (<), greater than (>), less than or equal to (< ), and greater than or equal to (> ). A linear equation has only one solution, but a linear inequality has an infinite number of solutions. When you write
✔ If a < b, then a + c < b + c (adding any number c).
✔ If a < b, then a – c < b – c (subtracting any number c).
✔ If a < b, then
✔ If a < b, then
✔ If a < b, then
✔ If a < b, then
✔ If
Notice that the direction of the inequality changes only when multiplying or dividing by a negative number or when reciprocating (flipping) fractions.
To solve a basic linear inequality, you first move all the variable terms to one side of the inequality and the numbers to the other. After you simplify the inequality down to a variable and a number, you can find out what values of the variable will make the inequality into a true statement. For example, to solve 3x + 4 > 11 – 4x, you add 4x