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3 years ago

Solve |x + 7| < 6

Mathematics

2 answers:

ASHA 777 [7]3 years ago

7 0

Answer:

A) –13 < x < –1

Step-by-step explanation:

|x + 7| < 6

Separate into possible cases

$x + 7 < 6 \\x +7 \geq 0$

Solve the inequalities

$x < - 1, x \geq -7\\x > -13, x < -7$

Find the intersections

$x = (-7, -1)\\x = (-13, -7)$

Find the union

Answer: A) –13 < x < –1

Vedmedyk [2.9K]3 years ago

4 0

Answer:

A) - 13 < x < - 1

Step-by-step explanation:

1) Rewrite the inequality without the absolute value.

$- 6 < x + 7 < 6$

2) Subtract 7 from the whole equation.

$- 6 - 7 < x < 6 - 7$

3) Simplify.

$- 13 < x < - 1$

Therefor, the answer is the first option.

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Choose the set of three sides that correctly matches each description. Each option will be used only once.

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Answers:

30-60-90 triangle = 4, 8, 4*sqrt(3)
45-45-90 triangle = 6, 6, 6*sqrt(2)
Right triangle, but not special right triangle = 6, 8, 10
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======================================================

Explanation:

The template for a 30-60-90 triangle is that the short leg x leads to hypotenuse 2x, and longer leg x*sqrt(3). If x = 4, then we get the sides 4, 8, 4*sqrt(3) which are the short leg, hypotenuse, and long leg in that order.

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The template for a 45-45-90 triangle is that we have x as the leg length and x*sqrt(2) as the hypotenuse. If x = 6, then we end up with the sides 6, 6, 6*sqrt(2)

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The triangle with sides 6, 8, 10 is a right triangle because a = 6, b = 8, c = 10 makes a^2+b^2 = c^2 true. I'm using the converse of the pythagorean theorem. We don't have a special right triangle because these sides do not fit the templates mentioned in the previous two sections above.

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The triangle with sides a = 7, b = 8, c = 10 on the other hand will not make a^2+b^2 = c^2 true. Note that a^2+b^2 = 7^2+8^2 = 113 which is not the same as 10^2 = 100. However, a triangle is possible because adding any two sides leads to a sum larger than the third side (triangle inequality theorem).

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We cannot form a triangle with sides 7, 12, 19 because the sides 7 and 12 add to 7+12 = 19 but that's not longer than the third side 19.

The triangle inequality theorem says that if you want to form a triangle with sides a,b,c then the following three inequalities must all be true

a+b > c

a+c > b

b+c > a

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