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The Appearance of a Polynomial Equation
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Factoring Trinomials of the Type ax 2 + bx + c
The Distance Formula
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Multiplication by 25
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Product Rule for Radicals
Solving Compound Linear Inequalities
Axis of Symmetry and Vertices
Multiplying Rational Expressions
Reducing Rational Expressions
Properties of Negative Exponents
Fractions
Numbers, Factors, and Reducing Fractions to Lowest Terms
Solving Quadratic Equations
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Example

Graph the system of inequalities.
x

x - 2y

≤ 0

> 2

 

Solution

Step 1 Solve the first inequality for y. Then graph the inequality.

The first inequality does not contain the variable y.

To graph the first inequality, x 0, first graph the corresponding equation, x = 0.

• This is a vertical line that passes through the x-axis at the point (0, 0); it is the y-axis.

For the inequality x 0, the inequality symbol is “”. This stands for “is less than or equal to.”

• To represent “equal to,” draw a solid line along the y-axis.

• To represent “less than,” shade the region to the left of the line. Each point in that region has an x-coordinate less than 0.

Step 2 Solve the second inequality for y. Then graph the inequality.

To solve x - 2y > 2 for y, do the following:

Subtract x from both sides. - 2y > - x + 2
Divide both sides by -2. Be sure to reverse the inequality symbol because you are dividing by a negative number.
Simplify.
To graph , first graph the equation .

• The y-intercept is (0, -1). Plot (0, -1).

• The slope is . To find a second point on the line, start at (0, -1) and move up 1 and right 2 to the point (2, 0). Plot (2, 0).

• Since the inequality symbol “<” does not contain “equal to,” draw a dotted line through (0, -1) and (2, 0).

• To represent “less than,” shade the region below the line.

Step 3 Shade the region where the two graphs overlap.

The solution is the region where the graphs overlap.

The solution of the system is the dark shaded region.

As a check, choose a point in the solution region.

For example, choose ( -1, -5).

To confirm that ( -1, -5) is a solution of the system, substitute -1 for x and -5 for y in each of the original inequalities and simplify.

First inequality Second inequality

x

Is -1

≤ 0

≤ 0 ? Yes

 

Is

Is

Is

x - 2y

-1 - 2(-5)

-1 + 10

9

> 2

> 2 ?

> 2 ?

> 2 ? Yes

 

Since ( 1, 5) satisfies each inequality, it is a solution of the system.