Solving Linear Inequalities
Graphing Linear Inequalities
Systems of Equations and Inequalities
Exponents and Exponential Functions
Polynomials and Factoring
100
The speed limit on the highway is 60 miles per hour
What is s<= 60 Reason: You can drive at a speed below 60 or you can drive at a speed of 60, but you cannot drive at a speed above 60.
100
On a number line, one variable is graphed to represent all possible solutions. An open circle is used for < or> and a closed circle is used for <= or >=.
What is the graph of an inequality?
100
Two or more linear equations in the same variables.
What is system of linear equations?
100
(7^3)(7^5)
What is 7^8 Reason: When you multiply exponents, you are really just addig them, so 7^3 times 7^5 is't 7^15, but rather, 7^8.
100
Find the sum: (2x^3-5x^2+x) + (x^3+2x^2-1)
What is 3x^3-3x^2+x-1? Reason: When you add or subtract polynomials, you do it based on the exponent. For instance, you couldn't add 5x^3 and 5x^2 because they have different exponents.
200
A child must be taller tha 48 inches to get on an amusement park ride.
What is h>=48 Reason: You can be taller than 48 inches or you could be exactly 48 inches, but you cannot be shorter tha 48 inches.
200
An ordered pair that satisfies each equation in the system.
What is a solution of linear equations?
200
6^7 times 3^6
What is no solution? Reason: When you multiply exponents, the base has to be the same.
200
(3a+4)(a-2)
What is 3a^2-2a-8? Reason: The easiest way to multiply two binomials is the FOIL method. FOIL stands for First Inside Outside Last. Basically, you do 3a times a first the 3a times -2, ad the 4 times a and 4 times -2. Once you've done that, you add like terms and voila!
300
x+4<5
What is x<1? Reason: As previously learned, the inverse operation of addition is subtraction, so you subtract 4 from both sides, leaving x<1.
300
Solve the system: x=y+3 2x-y=5
What is (2,-1)? Reason: There are three basic ways to solve this. The first way is by graphing both lines. The problem with graphing is that sometimes it is hard to tell where the two lies intersect, if the intersecting point isn't whole. In this case, there wouldn't be any problems because both coordinates are whole numbers. The second way is through substitution. The first step in substitution is getting one of the variables in one of the equations by itself. We ca skip this step in this system because x=y+3 already has x by itself. Now, we have to substitute y+3 for x in the other equation giving us, 2(y+3)-y=5. After applying the distibutive property, we get, 2y+6-y=5. Now, we must add like terms, which in this case is the 2y and the y. So, our equation becomes y+6=5. To get y by itself, we have to subtract 6 from both sides, and we get, y=-1. We still aren't done. Finally we have to find the value of x, so we substitute -1 for y in the first equation, add like terms, and we get x=2. The final way we could solve is through elimination. To do that, we must first put the first equation into standard form. To do that, we have to subtract y from both sides, and we get x-y=3. Now, we have to add or subtract the equations to get rid of one variable. In this case, because neither y have a coefficient before them, that is the easiest variable to use (x would require multiplying the entire first equatio by 2). Because they are both negative, we are going to subtract because subtracting a negative is like adding a positive. After we subtract, our equation becomes, -x=-2. Since -x=-2, we know that x=2. Now, we go back to either one of the equations, and substitute 2 for x. After you've done that, solve the equation. You should end up with -1 for y, but ALWAYS check your math!
300
8^10/8^4
What is 8^6? Reason: When you multiply two exponents, you are actually adding the exponents. When you divide two exponents, you are actually subtracting the exponents.
300
(x-y)(3x+4y)
What is 3x^2+xy-4y^2? Reason: Basically, you just follow the foil method of irst Inside Outside Last. The only issue is when you get to the xy part. Because you don't know what x or y is valued at, you can't simplify them any further than xy.
400
|d+4|>=3
What is d>=-1 or d<=-7 Reason: With problems that contain absolute value, you must solv e for the positive result (|d+4|>=3) and the negative (|d+4|>=-3) because since the absolute value is the distance from zero there are always two answers. The steps are the same, but the answers are different because one of the inequalities is negative. So you subtract 4 from both sides, in the first inequality, you get -1 and in the second, you get -7, but because you multiplied the inequality by -1 in order to get -3, the sign was reversed making it d<=-7.
400
Solve the system using substitution: y=4x-9 y=-8x+15
What is (2,-1) Reason: Because both equations have already been solved for the same variable, they equal each other, so the equation is 4x-9=-8x+15. Now, you solve it just like you would solve any equation with variables on both sides, you get the variable on one side and the numbers on the other. Once you've done that, you should get 12x=24, you divide both sides by 12 ad you get x=2. Now, you go back to either one of the equations, and substitute 2 in for x and solve. You should get -1.
400
Write without negative: 3^-2
What is 1/ 3^2? Reason: 3^-2 is actually the reciprocal of 3^2.
400
Solve by factoring: 2x^2+8x=0
What is (2x)(x+4) x=0 or x=-4 Reason: With factoring, you are trying to take out something that all the numbers in the equation (besides 0) share. In this case, that was 2x. Once you've done that, you solve for x.
500
-3|2- 5/4u|<=18
What is u<=-3 1/5 or u>=6 2/5 Reason: First, to get rid of the -3, you have to divide both sides of the inequality by -3. Then the inequality becomes 2- 5/4u>=6. To get rid of the 2, you subtract 2 from both sides because 2 is positive. Then, the inequality becomes -5/4u>=4. The, you multiply both sides by -4/5 because that is -5/4's reciprocal. Because you multiplied it by a negative, the sign reverses again. The inequality becomes u<= -3 1/5. Then you solve for the negative (-18), which switches the sign. You follow the same steps and end up with u>=6 2/5.
500
Factor: n^2+4n-5
What is (n-1) (n+5). Reason: When factoring x^2+bx+c, the two numbers are multiplied to equal c, and those numbers are also added to equal b. In this case, those numbers were -1 and 5.