Algebra Problem Set

Solution Preview

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1. Find the solution sets for the following: |5x - 7| > 12

Pretend for a moment that the problem was |5x - 7| = 12. This would mean that 5x - 7 = 12 or that 5x - 7 = -12. You would then solve both equations to get two answers for x.

However, the actual problem has a ">" instead of an "=". This means that 5x - 7 can equal numbers like 13, 14, 15, ... or it can equal numbers like -13, -14, -15....

We have two inequalities we need to solve: 5x - 7 > 12 (the first scenario) and 5x - 7 < -12 (the second scenario):

5x - 7 > 12 5x - 7 < -12
5x > 19 5x < -5
x > 19/5 x < -1

Therefore, x can be greater than 19/5 OR it can be less than -1.

2. Find the equation of the line satisfying the following:
The line with x-intercept 7 and y-intercept -5.

The x-intercept is the point where the line crosses the x-axis (and therefore y = 0). Similarly, x = 0 at the y-intercept. Therefore, the information in the problem tells us two of the points on the line: (7, 0) and (0, -5).

I think the easiest way to solve linear equations is to think of the slope-intercept form of the line and solve for the slope and the y-intercept (it is especially easy in this case, because they give us the y-intercept directly).

The slope-intercept equation of a line looks like this:

y = mx + b

where m is the slope, and b is the y-intercept.

We already know the y-intercept (it's -5), so now we need to find the slope.

You calculate the slope by taking two points on the line (which we have), and dividing the difference in the y-values by the difference in the x-values. So, in this case, the slope is:

(0) - (-5) = 5
(7) - (0) 7

The slope is 5/7.

Plugging the slope and the y-intercept into the equation gives us our answer:

y = (5/7)x - 5

3. Graph the functions L(x) = -3x + 17 and Q(x) = 2x2 + 7x - 22 on the same set of axes.

There are two ways of graphing functions: (1) you can find lots of points that satisfy the function, then plot those points and connect them, or (2) you can use the equation to find properties of the function (vertex, slope, intercepts), and use that information to make a graph.

Because it's easy to find a slope and intercept of a line, we'll use method 2 to graph L(x), and because it's more difficult to find similar information about a parabola, we'll use method 1 to graph Q(x).

L(x) = -3x + 17

From the equation, we know that the y-intercept is at 17. Put a dot at the point (0, 17). We also know that the slope is -3. Starting at the point (0, 17), move to the right 1 and down 3 and put another dot there. Keep moving right 1 and down 3 until you have enough points that you can connect in a line.

Q(x) = 2x2 + 7x - 22

For this, pick different values of x, then plug them into the equation. Once you get an answer (that will be your y-value), you can plot that point on the graph. When you find enough points, you'll be able to connect them into a parabola.

For example, look at x = 3:

Q(3) = 2(3)2 + 7(3) - 22
Q(3) = 2*9 + 21 - 22
Q(3) = 18 + 21 - 22
Q(3) = 18 - 1
Q(3) = 17

Therefore, one point on the graph is (3, 17).

If you find the y-values for lots of x's, you'll have multiple points to make a graph with:

x y
-7 27
-6 8
-5 -7
-4 -18
-3 -25
-2 -28
-1 -27
0 -22
1 -13
2 0
3 17
4 38
5 63
6 92
7 125

The graph looks like this:

4. Determine the x-intercepts of g(x) = x2 - x - 20

The x-intercepts are the points when y = 0 (i.e. when g(x) = 0). So, we want to solve the following:

0 = x2 - x - 20

To solve a quadratic equation like this, you can always use the quadratic formula

x = -b ± &#8730;( b2 - 4*a*c )
2a

where in this case, a = 1, b = -1, and c = -20.

However, if there is an easy way to factor the equation, you can do that too. (Keep in mind that you can't factor everything, but the quadratic equation always works, so if you're not good at factoring, just use the quadratic equation for everything.)

Looking at 0 = x2 - x - 20, I want to think of two numbers that we can multiply together to get -20 and that we can add together to get -1. Two such numbers are -5 and 4. This means that we can factor the equation as...

0 = x2 - x - 20
0 = (x - 5)(x + 4) (You should multiply this out to confirm that it equals 0 = x2 - x - 20)

When two things multiply together to equal 0, that means that the first one equals 0, the second one equals 0, or both of them equal 0. In this case, either x - 5 = 0 or x + 4 = 0. Solve both equations:

x - 5 = 0 x + 4 = 0
x = 5 x = -4

The equation g(x) = x2 - x - 20 equals 0 when x = 5 and when x = -4 (you can use the quadratic formula to get the same results). Therefore, the x-intercepts are (-4, 0) and (5, 0).

5. Determine the solution to the following system of equations:

2x - y - 4z = 9
x - 2y + z = 3
3x + y + z = -2

Basically, we want to add the equations together in such a way that variables cancel out, and we are left with a solution for x, y, or z. Then, we can plug that solution into one or more of the equations, and repeat the process until we get a solution for all of the variables.

First, multiply the second equation by -2 (if you multiply every term in an equation by one number, making sure that you multiply everything on both sides of the "=", this doesn't change any of the properties of the equation - i.e. it's still the same equation).

-2x + 4y - 2z = -6

Now, add that equation to the first one:

-2x + 4y - 2z = -6
+ 2x - y - 4z = 9
0x + 3y - 6z = 3

You get 3y - 6z = 3 (which is the ...