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2 years ago

Write the contrapositive and the converse of the following conditional

Mathematics

2 answers:

Lunna [17]2 years ago

6 0

The contrapositive statement are:

If the lake is frozen, then it isn't cold.
If Solomon is happy, then he isn't healthy.
If Tigist does not take a walk, then it will not rain

<h3>What is the converse statement?</h3>

The converse statement are:

If the late is frozen, then it is cold.
If Solomon is happy, then he is healthy.
If Tigist Tigist does not take a walk, then it will rain.

Note that the converse of a statement is created by the act of switching the hypothesis given and also the conclusion.

Therefore, The contrapositive statement are

If the lake is frozen, then it isn't cold.
If Solomon is happy, then he isn't healthy.
If Tigist does not take a walk, then it will not rain

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kenny6666 [7]2 years ago

4 0

The contrapositive statement is:

If the lake is frozen, then it isn't cold.

If Solomon is happy, then he isn't healthy.

If Tigist does not take a walk, then it will not rain.

We have given that the statements,

We have to determine the contrapositive and the converse of the following conditional statements.

<h3>What is the converse statement?</h3>

The converse statement is:

If the late is frozen, then it is cold.

If Solomon is happy, then he is healthy.

If Tigist Tigist does not take a walk, then it will rain.

Note that the converse of a statement is created by the act of switching the hypothesis is given and also the conclusion.

Therefore, The contrapositive statement is

If the lake is frozen, then it isn't cold.

If Solomon is happy, then he isn't healthy.

If Tigist does not take a walk, then it will not rain.

Learn more about contrapositive statements from

brainly.com/question/1713053

#SPJ1

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Opposite reciprocal slope lines or parallel or perpendicular

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Given,

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(a). The solutions are 0 and ⁸/₃.

(b). The solutions are 1 and ¹³/₃.

(c). The equation has no solution.

(d). The only solution is ²¹/₂₀.

(e). The equation has no solution.

<h3>Further explanation</h3>

These are the problems with the absolute value of a function.

For all real numbers x,

$\boxed{ \ |f(x)|=\left \{ {{f(x), for \ f(x) \geq 0} \atop {-f(x), for \ f(x) < 0}} \right. \ }$

<u>Problem (a)</u>

|4 – 3x| = |-4|

|4 – 3x| = 4

$\boxed{ \ 4 - 3x \geq 0 \ } \rightarrow \boxed{ \ 4\geq 3x \ } \rightarrow \boxed{ \ x\leq \frac{4}{3} \ }$

For 4 – 3x = 4

Subtract both sides by four.

-3x = 0

Divide both sides by -3.

x = 0

Since $\boxed{ \ 0\leq \frac{4}{3} \ }$ , x = 0 is a solution.

$\boxed{ \ 4 - 3x < 0 \ } \rightarrow \boxed{ \ 4 < 3x \ } \rightarrow \boxed{ \ x > \frac{4}{3} \ }$

For -(4 – 3x) = 4

-4 + 3x = 4

Add both sides by four.

3x = 8

Divide both sides by three.

$x = \frac{8}{3}$

Since $\boxed{ \ \frac{8}{3} > \frac{4}{3} \ }$ , $\boxed{ \ x = \frac{8}{3} \ }$ is a solution.

Hence, the solutions are $\boxed{ \ 0 \ and \ \frac{8}{3} \ }$

————————

<u>Problem (b)</u>

2|3x - 8| = 10

Divide both sides by two.

|3x - 8| = 5

$\boxed{ \ 3x - 8 \geq 0 \ } \rightarrow \boxed{ \ 3x\geq 8 \ } \rightarrow \boxed{ \ x\geq \frac{8}{3} \ }$

For 3x - 8 = 5

Add both sides by eight.

3x = 13

Divide both sides by three.

$x = \frac{13}{3}$

Since $\boxed{ \ \frac{13}{3} \geq \frac{4}{3} \ }$ , $\boxed{ \ x = \frac{13}{3} \ }$ is a solution.

$\boxed{ \ 3x - 8 < 0 \ } \rightarrow \boxed{ \ 3x < 8 \ } \rightarrow \boxed{ \ x < \frac{8}{3} \ }$

For -(3x – 8) = 5

-3x + 8 = 5

Subtract both sides by eight.

-3x = -3

Divide both sides by -3.

x = 1

Since $\boxed{ \ 1 < \frac{8}{3} \ }$ , $\boxed{ \ x = 1 \ }$ is a solution.

Hence, the solutions are $\boxed{ \ 1 \ and \ \frac{13}{3} \ }$

————————

<u>Problem (c)</u>

2x + |3x - 8| = -4

Subtracting both sides by 2x.

|3x - 8| = -2x – 4

$\boxed{ \ 3x - 8 \geq 0 \ } \rightarrow \boxed{ \ 3x\geq 8 \ } \rightarrow \boxed{ \ x\geq \frac{8}{3} \ }$

For 3x – 8 = -2x – 4

3x + 2x = 8 – 4

5x = 4

$x = \frac{4}{5}$

Since $\boxed{ \ \frac{4}{5} \ngeq \frac{8}{3} \ }$ , $\boxed{ \ x = \frac{4}{5} \ }$ is not a solution.

$\boxed{ \ 3x - 8 < 0 \ } \rightarrow \boxed{ \ 3x < 8 \ } \rightarrow \boxed{ \ x < \frac{8}{3} \ }$

For -(3x - 8) = -2x – 4

-3x + 8 = -2x – 4

2x – 3x = -8 – 4

-x = -12

x = 12

Since $\boxed{ \ 12 \nless \frac{8}{3} \ }$ , $\boxed{ \ x = 12 \ }$ is not a solution.

Hence, the equation has no solution.

————————

<u>Problem (d)</u>

5|2x - 3| = 2|3 - 5x|

Let’s take the square of both sides. Then,

[5(2x - 3)]² = [2(3 - 5x)]²

(10x – 15)² = (6 – 10x)²

(10x - 15)² - (6 - 10x)² = 0

According to this formula $\boxed{ \ a^2 - b^2 = (a + b)(a - b) \ }$

$[(10x - 15) + (6 - 10x)][(10x - 15) - (6 - 10x)]] = 0$

(-9)(20x - 21) = 0

Dividing both sides by -9.

20x - 21 = 0

20x = 21

$x = \frac{21}{20}$

The only solution is $\boxed{ \ \frac{21}{20} \ }$

————————

<u>Problem (e)</u>

2x + |8 - 3x| = |x - 4|

We need to separate into four cases since we don’t know whether 8 – 3x and x – 4 are positive or negative. We cannot square both sides because there is a function of 2x.

8 – 3x is positive (or 8 - 3x > 0)
x – 4 is positive (or x - 4 > 0)

2x + 8 – 3x = x – 4

8 – x = x – 4

-2x = -12

x = 6

Substitute x = 6 into 8 – 3x ⇒ 8 – 3(6) < 0, it doesn’t work, even though when we substitute x = 6 into x - 4 it does work.

8 – 3x is positive (or 8 - 3x > 0)
x – 4 is negative (or x - 4 < 0)

2x + 8 – 3x = -(x – 4)

8 – x = -x + 4

x – x = = 4 - 8

It cannot be determined.

8 – 3x is negative (or 8 - 3x < 0)
x – 4 is positive. (or x - 4 > 0)

2x + (-(8 – 3x)) = x – 4

2x – 8 + 3x = x - 4

5x – x = 8 – 4

4x = 4

x = 1

Substitute x = 1 into 8 - 3x, $\boxed{ \ 8 - 3(1) \nless 0 \ }$ , it doesn’t work. Likewise, when we substitute x = 1 into x – 4, $\boxed{ \ 1 - 4 \not> 0 \ }$

8 – 3x is negative (or 8 - 3x < 0)
x – 4 is negative (or x - 4 < 0)

2x + (-(8 – 3x)) = -(x – 4)

2x – 8 + 3x = -x + 4

5x + x = 8 – 4

6x = 4

$\boxed{ \ x=\frac{4}{6} \rightarrow x = \frac{2}{3} \ }$

Substitute $x = \frac{2}{3} \ into \ 8-3x$ , $\boxed{ \ 8 - 3 \bigg(\frac{2}{3}\bigg) \not< 0 \ }$ , it doesn’t work. Even though when we substitute $x = \frac{2}{3} \ into \ x-4$ , $\boxed{ \ \bigg(\frac{2}{3}\bigg) - 4 < 0 \ }$ it does work.

Hence, the equation has no solution.

<h3>Learn more</h3>

The inverse of a function brainly.com/question/3225044
The piecewise-defined functions brainly.com/question/9590016
The composite function brainly.com/question/1691598

Keywords: hitunglah nilai x, the equation, absolute value of the function, has no solution, case, the only solution

5 0

3 years ago

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