Match the equation with its solution.
3i + 2i -7 =18
3(2i - 7) = 15
3i + 5 = 2i -7
3i + 5 = 3i + 7
3(2i + 7) = 6i + 21
choose one from the table to match the above equation.
i = -12
i = 5
i = 6
i = 7
i = 11
i = 2
The equation and its solution are:
3i + 2i -7 =18 (i = 7)
3(2i - 7) = 15
The equation and its solution are:
3(2i - 7) = 15 (i = 11)
3i + 5 = 2i -7
The equation and its solution are:
3i + 5 = 2i -7 (i = -12)
3i + 5 = 3i + 7
The equation and its solution are:
3i + 5 = 3i + 7 (No solution)
3(2i + 7) = 6i + 21
The equation and its solution are:
3(2i + 7) = 6i + 21 (i = 2)
Solve the equation justifying with the correct reasoning: 2(x+8)=2x +8
To solve the equation 2(x + 8) = 2x + 8, we need to simplify both sides of the equation.
First, we distribute the 2 on the left side:
2 * x + 2 * 8 = 2x + 8
2x + 16 = 2x + 8
Next, we want to isolate the variable x. By subtracting 2x from both sides, we can get rid of the 2x term on the right side:
2x - 2x + 16 = 2x - 2x + 8
16 = 8
However, we can see that 16 does not equal 8.
Therefore, the equation 2(x + 8) = 2x + 8 has no solution.
Match the description of the one variable equation with the number of solutions it will have. (One Solution, No Solution, Infinite Solutions)
1. x - 7 = 7 - x
2. 3(x + 5) = 3x + 5
3. 10 - x = 25
4. 2(x + 4) = 2x + 5 + 3
The descriptions of the one-variable equations and their number of solutions are as follows:
1. x - 7 = 7 - x (One Solution)
2. 3(x + 5) = 3x + 5 (One Solution)
3. 10 - x = 25 (No Solution)
4. 2(x + 4) = 2x + 5 + 3 (Infinite Solutions)
A student wants to purchase some new school supplies. He wants to buy a calculator that costs $24 and some notebooks for school. Each notebook costs $2. The student only has $37 to spend.
Let n represent the number of notebooks that he buys.
Which inequality describes this scenario?
A. 37 greater than equal to 2n + 24
B. 37 less than 2n + 24
C. 37 greater than 2n + 24
D. 24n + 2 greater than equal to 37
The inequality that describes this scenario is:
B. 37 less than 2n + 24
Since the student has $37 to spend, the total amount he spends on notebooks and calculator (2n + 24) must be less than $37.
Solve for b in the following formula: A = 1/2(a + b)
To solve for b in the formula A = 1/2(a + b), we can follow the following steps:
1. Multiply both sides of the equation by 2 to remove the fraction:
2A = a + b
2. Subtract a from both sides:
2A - a = b
Therefore, the solution for b is:
b = 2A - a
Graph the solution set for the inequality: -3x + 1 less than or equal to -47
To graph the solution set for the inequality -3x + 1 ≤ -47, we can follow these steps:
1. Start by graphing the equation -3x + 1 = -47. To do this, first, subtract 1 from both sides of the equation:
-3x = -48
2. Divide both sides of the equation by -3 to isolate x:
x = 16
3. Now, in order to graph the inequality -3x + 1 ≤ -47, we need to determine whether to shade the area to the left or right of the line x = 16.
Since the inequality is less than or equal to, we need to include the points on the line as part of the solution set. Thus, we will shade the area to the left of the line.
Therefore, we draw a vertical dashed line at x = 16 and shade the area to the left of the line.
The graph should look like this:
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-47◄══════════════════════►
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Note: The dashed line indicates that the points on the line (x = 16) are included in the solution set.
A student claims the graph below represents the solution to the inequality -4 less than x with this graph. What was the students mistake. Student graph is open circle -4 with line to the left of the -4.
The student's mistake is that they plotted an open circle at -4 and shaded the area to the left of the -4. However, the inequality -4 < x (or x > -4) represents all values of x that are greater than -4, not including -4 itself.
In order to correctly graph the inequality -4 < x, the student should have drawn an open circle at -4 and shaded the area to the right of the -4. This indicates that the solution set includes all values of x greater than -4, but does not include -4 itself.
Correcting the student's mistake, the graph should look like this:
─●─►───────────
In this corrected graph, the open circle at -4 indicates that -4 is not included in the solution set, and the arrow to the right indicates that the solution set extends to positive infinity.
Grandma is removing weeds from her garden. She estimates that there are 250 weeds in the garden at the moment, and she can remove weeds at a rate of 5 per minute. At this rate how many minutes (m) will it take her to only have 30 weeds remaining in the garden?
What is the equation?
What is the solution?
Meaning of the solution in words?
The equation that represents the situation is:
250 - 5m = 30
In this equation, "m" represents the number of minutes and it is being multiplied by the removal rate of 5 weeds per minute. The left side of the equation represents the initial number of weeds in the garden (250), decreasing by 5 weeds for each minute spent removing them.
To find the solution and determine the number of minutes it will take for Grandma to have 30 weeds remaining in the garden, we can solve the equation:
250 - 5m = 30
Subtract 250 from both sides:
-5m = 30 - 250
-5m = -220
Divide both sides by -5:
m = (-220) / (-5)
m = 44
The solution to the equation is m = 44.
In words, it will take Grandma 44 minutes to remove enough weeds from the garden to have only 30 remaining.
The Celsius to Fahrenheit formula: F=9/5 C+32(9/5 is a fraction)
Solve this formula for C. Check all answers that are true.
To solve the Celsius to Fahrenheit formula F = (9/5)C + 32 for C, we need to isolate C on one side of the equation.
Step 1: Subtract 32 from both sides of the equation to get rid of the constant term:
F - 32 = (9/5)C
Step 2: Multiply both sides of the equation by the reciprocal of (9/5) which is (5/9):
(5/9)(F - 32) = C
So the solution for C is C = (5/9)(F - 32).
Now, let's check the given answer choices to determine which ones are true.
A. C = (F - 32)/(9/5)
To check if this answer choice is true, we need to simplify it:
C = (F - 32)/(9/5)
C = (F - 32) * (5/9)
C = (5/9)F - (5/9)32
C = (5/9)F - 160/9
This answer choice is not true. It does not match the solution we found, C = (5/9)(F - 32).
B. C = (9/5)(F - 32)
To check if this answer choice is true, we can simplify it:
C = (9/5)(F - 32)
C = (9/5)F - (9/5)32
C = (9/5)F - 288/5
This answer choice is not true. It does not match the solution we found, C = (5/9)(F - 32).
Therefore, none of the given answer choices are true. The correct solution for C is C = (5/9)(F - 32).
Izzie is trying to solve this inequality: 3(t+1)−4t≥−5
After she solves it, she shows it to you, and she asks you if she did it correctly.
This is the work she completed:
3t+3−4t≥−5
−t+3≥−5
−t≥−8
t≥8
Izzie's work appears to have a mistake. Let's go through the solution step by step to determine the error.
Starting with the original inequality: 3(t+1) - 4t ≥ -5
Distributing the 3 on the left side:
3t + 3 - 4t ≥ -5
Simplifying the left side:
-t + 3 ≥ -5
This part is correct.
Now, to isolate t, Izzie needs to subtract 3 from both sides:
-t + 3 - 3 ≥ -5 - 3
-t ≥ -8
However, the mistake occurs when solving for t. When multiplying or dividing by a negative number, the direction of the inequality flips. In this case, since we are dividing by -1, the inequality sign should flip.
To correct the mistake:
Since -t ≥ -8, when we divide both sides by -1, we must flip the inequality sign:
t ≤ 8
Therefore, the correct solution is t ≤ 8, not t ≥ 8 as Izzie initially wrote.
Which graph represents the solutions to this inequality: 5n−3<3(n+1)
To determine which graph represents the solutions to the inequality 5n - 3 < 3(n + 1), we can solve the inequality and then analyze the graph of the solution set.
First, let's solve the inequality:
5n - 3 < 3(n + 1)
Distribute on the right side:
5n - 3 < 3n + 3
Subtract 3n from both sides:
2n - 3 < 3
Add 3 to both sides:
2n < 6
Divide by 2 on both sides:
n < 3
The inequality solution is n < 3. Now, let's analyze the graph of the solution set.
The graph should show a shaded area to the left of 3 (exclusive) on a number line because the values of n that satisfy the inequality are less than 3.
Out of the given answer choices, the correct graph that represents the solutions to the inequality is:
───◄3───────►
In this graph, there is an open circle at 3 (indicating that 3 is not included in the solution set), and the shaded area is to the left of 3.
Solve the equation and inequality for x.
Part A: Solve the equation showing all steps.
ax−5=18
Part B: Solve the inequality showing all steps.
ax−5>18
Answer these Questions:
1. Describe the number line set up that would show all of the solutions for the inequality in Part B.
2. State the difference between solving an equation and an inequality.
Part A: To solve the equation ax - 5 = 18, we can follow these steps:
1. Add 5 to both sides of the equation to isolate the term with "x":
ax - 5 + 5 = 18 + 5
ax = 23
2. To solve for x, divide both sides of the equation by "a":
(ax) / a = 23 / a
x = 23/a
So, the solution to the equation ax - 5 = 18 is x = 23/a.
Part B: To solve the inequality ax - 5 > 18, we can follow these steps:
1. Add 5 to both sides to isolate the term with "x":
ax - 5 + 5 > 18 + 5
ax > 23
2. To solve for x, divide both sides of the inequality by "a". It's important to note that if a is negative, the inequality sign should be flipped:
(ax) / a > 23 / a
x > 23/a (if a > 0)
x < 23/a (if a < 0)
So, the solution to the inequality ax - 5 > 18, depending on the sign of "a", is either x > 23/a or x < 23/a.
Answering the Questions:
1. To represent all solutions on a number line for the inequality ax - 5 > 18, we would draw an open circle at 23/a and shade the area to the right of it. This indicates that any value greater than 23/a would satisfy the inequality.
●───►
2. The main difference between solving an equation and an inequality is that when solving an equation, we are finding a specific value for the variable that makes the equation true. In contrast, when solving an inequality, we are finding a range of values for the variable that satisfy the inequality. Equations have one unique solution, while inequalities can have an infinite number of solutions.