Solve:
20. [tex]$3 x-4=11$[/tex]
a. 5
b. 3
c. -5
d. 4
21. What is the solution of
[tex]$\frac{2 x+1}{3}=5$[/tex]
a. 7
b. 6
c. 5
d. 4
22. On a number line, the inequality [tex]$x\ \textgreater \ -1$[/tex] is represented by
a. Open circle at -1, shading to the right
b. Closed circle at -1, shading to the left
c. Open circle at -1, shading to the left
d. Closed circle at -1, shading to the right
Answer (2)
Solve 3 x − 4 = 11 by adding 4 to both sides and dividing by 3, resulting in x = 5 .
Solve 3 2 x + 1 = 5 by multiplying both sides by 3, subtracting 1, and dividing by 2, resulting in x = 7 .
Represent -1"> x > − 1 on a number line with an open circle at -1 and shading to the right.
The solutions are x = 5 , x = 7 , and an open circle at -1 shading to the right, corresponding to options a, a, and a, respectively. 5 , 7 , Open circle at -1, shading to the right
Explanation
Problem Analysis We are given three separate math problems to solve:
Solve the equation 3 x − 4 = 11 .
Solve the equation 3 2 x + 1 = 5 .
Determine the correct representation of the inequality -1"> x > − 1 on a number line.
Solving the First Equation Let's solve the first equation, 3 x − 4 = 11 . To isolate x , we first add 4 to both sides of the equation:
3 x − 4 + 4 = 11 + 4
3 x = 15
Now, we divide both sides by 3:
3 3 x = 3 15
x = 5
So, the solution to the first equation is x = 5 , which corresponds to option a.
Solving the Second Equation Next, let's solve the second equation, 3 2 x + 1 = 5 . To isolate x , we first multiply both sides of the equation by 3:
3 × 3 2 x + 1 = 5 × 3
2 x + 1 = 15
Now, we subtract 1 from both sides:
2 x + 1 − 1 = 15 − 1
2 x = 14
Finally, we divide both sides by 2:
2 2 x = 2 14
x = 7
So, the solution to the second equation is x = 7 , which corresponds to option a.
Analyzing the Inequality Now, let's analyze the third problem. The inequality -1"> x > − 1 means that x can be any number greater than -1, but it cannot be equal to -1. On a number line, this is represented by an open circle at -1 (to indicate that -1 is not included) and shading to the right (to indicate that all numbers greater than -1 are included). This corresponds to option a.
Final Answer Therefore, the answers are:
a. 5
a. 7
a. Open circle at -1, shading to the right
Examples
Understanding how to solve simple equations and inequalities is crucial in many real-life situations. For example, if you are trying to determine how many hours you need to work to earn a certain amount of money, you might set up an equation like 15 x = 300 , where x is the number of hours and 15 i syo u r h o u r l y w a g e . S o l v in g t hi se q u a t i o n g i v esyo u t h e n u mb ero f h o u rsyo u n ee d t o w or k . S imi l a r l y , i f yo u w an tt o kn o wt h e minim u m scoreyo u n ee d o na t es tt o g e t a cer t ain g r a d e , yo u mi g h t u se anin e q u a l i t y . F or in s t an ce , i f yo u n ee d ana v er a g eo f a tl e a s t 80 , an d yo u ′ v e a l re a d yscore d 70 an d 90 o n tw o t es t s , t h e in e q u a l i t yco u l d b e \frac{70 + 90 + x}{3} \geq 80 , w h ere x$ is the score you need on the third test. Solving inequalities helps you determine the range of values that satisfy certain conditions, which is very useful in planning and decision-making.
The solutions are x = 5 for the first equation, x = 7 for the second equation, and an open circle at -1 with shading to the right for the inequality. Therefore, the correct options are a, a, and a, respectively.
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