Practice Makes Perfect Algebra - Carolyn Wheater [9]
To solve an inequality that involves an absolute value, first isolate the absolute value and then rewrite the inequality as a compound inequality. If the absolute value is greater than the expression on the other side, it will become an or inequality. Absolute values less than an expression translate to an and inequality.
|3x + 2| > 5 becomes −5 > 3x + 2 or 3x + 2 > 5
|3 − 7x| ≤ 10 becomes −10 ≤ 3x − 7x ≤ 10
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EXERCISE 3.3
Solve each inequality and graph the solution set on a number line.
1. |2x − 7| < 9
2. |3x + 5| ≥ 17
3. |3x + 5| + 4 > 36
4. |4 − 5x| − 3 ≤ 2
5. 2|7x − 12| − 5 > 13
6. −4|9 − x| + 7 < − 17
7. 2 −|11x + 5| ≤ − 36
8. 4 |17 − 2x| −9 ≥ 19
9. |6x − 11| < − x + 3
10. |13 − 5x| ≥ 2x − 1
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Coordinate graphing
The graph of an equation in two variables gives a picture of all the pairs of numbers that balance the equation. Studying the graph will help you understand the relationship between the variables and can sometimes help you find the solution of an equation.
The coordinate plane
The cartesian coordinate system, named for René Descartes, is a rectangular coordinate system that locates every point in the plane by an ordered pair of numbers(x, y). The x-coordinate indicates horizontal movement, and the y-coordinate vertical movement. Movement begins from a point(0, 0), called the origin, where two number lines, one horizontal and one vertical, intersect. The horizontal number line is the x-axis and the vertical is the y-axis. Positive x-coordinates are to the right of the origin, and negative x-coordinates to the left. A y-coordinate that is positive is above the x-axis, and a negative y-coordinate is below(see Figure 4.1).
Figure 4.1 The coordinate plane divided into four quadrants.
The x- and y-axes divide the plane into four quadrants. The first quadrant is the section in which both the x- and y-coordinates are positive, and the numbering of the quadrants goes counterclockwise.
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EXERCISE 4.1
Plot each point on the coordinate plane.
1. A(−4, 2)
2. B(8, −3)
3. C(0, 5)
4. D(−2, −6)
5. E(−5, 0)
Tell which quadrant contains the point.
6.(−4, 5)
7.(3, 2)
8.(4, −1)
9.(−2, −2)
10.(3, −4)
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Distance
The distance between two points can be calculated by means of the distance formula . The formula is an application of the pythagorean theorem, in which the difference of the x-coordinates gives the length of one leg of a right triangle, and the difference of the y-coordinates the length of the other. The distance between(x1, y1) and(x2, y2) is the hypotenuse of the right triangle. If the two points fall on a vertical line or on a horizontal line, the distance will simply be the difference in the coordinates that don’t match.
The distance between the points(4, −1) and(0, 2) is
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EXERCISE 4.2
Find the distance between the given points.
1.(4, 5) and(7, −4)
2.(6, 2) and(7, 6)
3.(−7, −1) and(−5, −6)
4.(5, 3) and(8, −2)
5.(−4, 2) and(3, 2)
Given the distance between the two points, find the possible values for the missing coordinate.
6.(a, −2) and(7, 2) are 5 units apart.
7.(−1, 3) and(4, d) are 13 units apart.
8.(8, −6) and(c, −6) are 7 units apart.
9.(2, b) and(2, −1) are 9 units apart.
10.(a, a) and(0, 0) are units apart.
Midpoints
The midpoint of the segment that connects(x1, y1) and(x2, y2) can be found by averaging the x-coordinates and averaging the y-coordinates.
The midpoint of the segment connecting(4, −1) and(0, 2) is
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EXERCISE 4.3
Find the midpoint of the segment with the given end points.
1.(2, 3) and(5, 8)
2.(−3, 1) and(−1, 8)
3.(−5, −3) and(−1, −1)
4.(8, 0) and(0, 8)
5.(0, −2) and(4, −4)
Given the midpoint