Practice Makes Perfect Algebra - Carolyn Wheater [11]
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EXERCISE 4.7
Identify each line as horizontal, vertical, or oblique.
1. x = − 3
2. y = 4
3. 2x + 8 = 0
4. y − 4x = 0
5. y + 1 = 4
Graph each equation.
6. y = −3
7. x = 2
8. y = 5
9. x = −1
10. 5y − 18 = 2
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Graphing linear inequalities
Linear inequalities can also be graphed on the coordinate plane. Begin by graphing the line that would result if the inequality sign were replaced with an equal sign. If the inequality is ≥ or ≤, use a solid line. For > or <, use a dotted line. Test a point on one side of the line in the inequality; the origin is often a convenient choice. If the result is true, shade that side of the line; if not, shade the other side(see Figure 4.3).
Figure 4.3 Graph of a linear inequality in two variables.
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EXERCISE 4.8
Graph each inequality and indicate the solution set by shading.
1. y ≤ 2x − 5
2. y > 5x − 4
3. y < x
5. y ≤ − 2x + 5
6. 3x − y > 0
7. 8y − 3x < −4
8. x − y ≥ −5
9. 2x − y ≤ −2
10. 2x − 3y ≥ −15
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Graphing absolute value equations
Many equations involving absolute value have graphs that are composed of two linear segments that have opposite slopes. The graph of y = |x|, for example, is made up of the graph of y = x when −x < 0 and the graph of y = x when x ≥ 0. The result is a V-shaped graph as shown in Figure 4.4.
When you start to build a table of values to graph an absolute value equation, first find the value of x that will make the expression in the absolute value signs equal 0. That will be the point of the V. Choose a few values below it and a few above it to fill out your table.
Figure 4.4 Graph of the absolute value equation.
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EXERCISE 4.9
Construct a table of values and graph the equation.
1. y = | x − 4|
2. y = | x | − 3
3. y = 2| x | + 1
4. y = 2| x + 1 |
5. y = −3| x |
8. y = | x − 2 | + 1
9. y = | x + 5 | −6
10. y = −2| x + 3 | −4
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Writing linear equations
It is sometimes necessary to determine the equation that describes a graph either by looking at the graph itself or by using information about the graph.
Slope and y-intercept
If the slope and y-intercept of the line are known or can be read from the graph, the equation can be determined easily by using the y = mx + b form. Replace m with the slope and b with the y-intercept.
Point and slope
If the slope is known and a point on the line other than the y-intercept is known, the equation can be found by using point-slope form: y − y1 = m(x − x1). Replace m with the slope, and replace x1 and y1 with the coordinates of the known point. Distribute and simplify to put the equation in y = mx + b form.
Two points
If two points on the line are known, the slope can be calculated using the slope formula. Once the slope is found, you can use the point-slope form and fill in the slope and either one of the two points.
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EXERCISE 4.10
Write the equation of the line described.
1. Slope = 3 and y-intercept(0, 8)
2. Slope = −5 and y-intercept(0, 2)
3. Slope = and y-intercept(0, 6)
4. Slope = 4 and passing through the point(3, 7)
5. Slope = and passing through the point(4, 3)
6. Slope = and passing through the point(−4, −1)
7. Passing through the points(0, 3) and(2, 7)
8. Passing through the points(3, 4) and(9, 8)
9. Passing through the points(0, −3) and(3, 1)
10. Passing through the points(2, 3) and(8, −6)
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Parallel and perpendicular lines
Parallel lines have the same slope. Perpendicular lines have slopes that multiply to −1, that is, slopes that are negative reciprocals. To find the equation of a line parallel to or perpendicular to a given line, first determine the slope of the given line. Be sure the equation is in slope-intercept form before trying to determine the slope. Use the same slope for a parallel line or the negative reciprocal for a perpendicular line, along with the given point, in point-slope form.
To find a line parallel to y = 3x −7 that passes through the point(4, −1), use