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enyata [817]
3 years ago
8

Find the solution set of this inequality|10x+20| ≤10

Mathematics
1 answer:
Pavlova-9 [17]3 years ago
7 0

Answer:

solution is

[-3,-1]

Step-by-step explanation:

we are given

|10x+20|\leq 10

Firstly, we will find critical values

so, let's assume it is equal

|10x+20|= 10

now, we can break absolute sign

For |10x+20|= -(10x+20):

-(10x+20)= 10

we can solve for x

-10x-20= 10

Add both sides by 20

-10x-20+20= 10+20

-10x= 30

Divide both sides by -10

and we get

x=-3

For |10x+20|= (10x+20):

(10x+20)= 10

we can solve for x

10x+20= 10

Subtract both sides by 20

10x+20-20= 10-20

10x= -10

Divide both sides by 10

and we get

x=-1

so, critical values are

x=-3

x=-1

now, we can draw a number line and locate these values

and then we can check inequality on each intervals

For (-\infty,-3):

We can select any random value from this interval and plug that in inequality

and we get

we can plug x=-5

|10\times -5+20|\leq 10

|-50+20|\leq 10

30\leq 10

so, this is FALSE

For [-3,-1]:

We can select any random value from this interval and plug that in inequality

and we get

we can plug x=-2

|10\times -2+20|\leq 10

|-20+20|\leq 10

0\leq 10

so, this is TRUE

For (-1,\infty):

We can select any random value from this interval and plug that in inequality

and we get

we can plug x=0

|10\times 0+20|\leq 10

|0+20|\leq 10

20\leq 10

so, this is FALSE

so, solution is

[-3,-1]

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The system of inequalities is the following:

i) <span>y ≤ –0.75x
ii)</span><span>y ≤ 3x – 2

since </span>0.75= \frac{75}{100}= \frac{3}{4}, we can write the system again as 

i) y \leq - \frac{3}{4}x
ii) y  \leq 3x-2

Whenever we are asked to sketch the solution of a system of linear  inequalities, we:

1. Draw the lines
2. Color the regions of the inequalities.
3. The solution is the region colored twice.


A.

to draw the line y =- \frac{3}{4}x

consider the points: (-4, 1) and (0, 0), or any 2 other points (x,y) for which y =- \frac{3}{4}x hold.

since we have an "smaller or equal to" inequality, the line is a solid line (not dashed, or dotted).

In order to find out which region of the line to color, consider a point not on the line, for example P(1, 1), which is clearly in the upper region of the line.

For (x, y)=(1, 1) the inequality y  \leq - \frac{3}{4}x, does not hold because 

1 \leq - \frac{3}{4}*1= -\frac{3}{4} is not true,

this means that the solution is the region of the line not containing (1, 1), as shown in picture 1.


B.
similarly, to draw the solution of inequality ii) y ≤ 3x – 2, 

we first draw the line y=3x-2, using the points (0, -2) and (2, 4), or any other 2 points (x,y) for which y=3x-2 holds.

after we draw the line, we can check the point P(1, 7) which clearly is above the line y=3x-2.

for (x, y) = (1, 7), the inequality y ≤ 3x – 2 does not hold

because 7 is not ≤ 3*1-2=1, so the region we color is the one not containing P(1, 7), as shown in picture 2.


The solution of the system is the region colored with both colors, the solid lines included. Check picture 3.

the lines intersect at (0.533, -0.4) because:

–0.75x=3x-2
-0.75x-3x=-2
-3.75x=-2, that is x= -2/(-3.75)=0.533

for x=0.533, y=3x-2=3(0.533)-2=-0.4

Answer: Picture 3, the half-lines included. So the graph is in the 3rd and 4th Quadrants

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If a polynomial is divided by a binomial and the remainder is zero, what does that tell you about the relationship between the b
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Step-by-step explanation:

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