Graph The Inequality 3x + Y ≥ 2x + 3
Graph The Inequality 3x Y 2x 3abcd2 Write The Equatio
1. Graph the inequality. 3(x – y) + 2x
2. Write the equation of the line with x-intercept (–4, 0) and undefined slope. Write your results in slope-intercept form, if possible.
a. y = –4
b. x = –4
c. y = –4x
d. x = 0
3. Determine which two equations represent parallel lines.
(a) y = –7x + 2
(b) y = 7x + 2
(c) y = x + 2
(d) y = –7x + 6
a. (a) and (b)
b. (b) and (c)
c. (a) and (c)
d. (a) and (d)
4. The inventor of a new product believes that the cost of producing the product is given by the function C(x) = 3.2x + 5,000. How much does it cost to produce 6,000 units of his invention?
a. $300
b. $3,000
c. $24,200
d. $30,200
5. Rewrite the equation 2x – 3y = –6 as a function of x.
a. f(x) = x + 2
b. f(x) = x - 3
c. f(x) = x - 6
d. f(x) = (2/3)x + 2
6. A copier was purchased by a company for $9,500. After 2 years, it is estimated that the value of the copier will be $8,300. If the value in dollars V and the time t in years are related by a linear equation, find the equation that relates V and t.
a. V = –600t + 9,500
b. V = –600t + 9,500
c. V = –600t + 9,500
d. V = –600t + 9,500
7. Determine which two equations represent perpendicular lines.
(a) y = x – 5
(b) y = 5x – 1
(c) y = -x + 3
(d) y = x – 2
a. (a) and (b)
b. (b) and (c)
c. (a) and (c)
d. (c) and (d)
8. One day, the temperature at 7:00 A.M. was 41°F, and by 3:00 P.M., the temperature was 57°F. What was the hourly rate of temperature change?
a. 4°F/h
b. 3°F/h
c. 2°F/h
d. 1°F/h
9. A line passing through (–9, –4) and (10, y) is perpendicular to a line with slope –1/2. Find the value of y.
a. 5
b. 7
c. 8
d. 9
10. Are the following lines parallel, perpendicular, or neither?
L1: through (9, 2) and (3, –8); L2: through (–3, 5) and (5, –1)
a. Parallel
b. Perpendicular
c. Neither
11. Write the equation of the line with slope –6 and y-intercept (0, 9).
a. y = –6x + 9
b. 9x – 6y = 0
c. y = –6x + 9
d. –6y = –54 + 6x
12. Find the slope of any line perpendicular to the line through points (5, 12) and (6, 2).
a. –10
b. 10
c. –1/10
d. 1/10
13. Find the y-intercept of the line represented by the following equation: 7x – y = –7
a. (0, 7)
b. (7, 0)
c. (0, –7)
d. (–7, 0)
14. You have at least $50 in change consisting of nickels and pennies. Write an inequality that shows the different number of coins in your piggy bank.
a. 5n + p ≥ 50
b. 5n + p ≤ 50
c. 5n + p ≥ 50
d. 5n + p ≤ 50
15. Given f(x) = –5x + 3, find f(a – 3).
a. –5a + 18
b. –5a + 3
c. –5a + 15
d. –5a + 0
16. Match the graph with one of the equations.
[Graph images omitted for context]
Paper For Above instruction
This comprehensive analysis explores various fundamental concepts in algebra, including graphing inequalities, writing equations of lines, determining parallel and perpendicular lines, and applying these principles to real-world problems. The discussion begins with the graphical representation of inequalities, illustrating how to plot solutions for inequalities such as 3(x – y) + 2x
Next, the focus shifts to identifying equations of lines based on specific conditions, such as the x-intercept at (–4, 0) with an undefined slope. Recognizing that an undefined slope corresponds to a vertical line leads to the equation x = –4. This type of problem underscores the importance of slope and intercept concepts in deriving line equations.
Further, the comparison of slopes facilitates the understanding of parallel and perpendicular lines. For instance, lines with identical slopes are parallel, while perpendicular lines have slopes that are negative reciprocals. Identifying these relationships involves analyzing given equations, such as y = –7x + 2 and y = 7x + 2 for parallel lines, and lines like y = x – 5 and y = –x + 3 for perpendicular relationships.
The application of linear equations to solve real-world problems is exemplified through cost functions. For example, calculating the cost to produce 6,000 units based on the function C(x) = 3.2x + 5,000 demonstrates the integration of algebraic models in economic contexts.
Additionally, rewriting standard forms of equations into slope-intercept form reveals insights into the behavior and characteristics of lines. Transformations of equations such as 2x - 3y = –6 into functions of x assist in understanding the relationship between variables and facilitate graphing techniques.
Linear relationships in various contexts, including depreciation of assets and temperature changes over time, are examined via deriving and interpreting line equations. For instance, modeling the value decrease of a copier over two years involves finding a linear function based on initial value and estimated depreciation.
Furthermore, determining the nature of line relationships—whether parallel or perpendicular—requires calculations of slopes from given points. The analysis extends to computing slopes, including the perpendicular slope to a line passing through specific points, illustrating the geometric principles underlying algebraic concepts.
The exploration concludes with practical applications such as writing coin inequalities and evaluating functions at certain points. These exercises reinforce understanding of variables, inequalities, and functional notation, essential components of algebraic literacy.
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