Solving & Graphing Inequalities And Equations

Hey Plastik Magazine readers! Let's dive into some math problems today. We're going to solve and graph a bunch of inequalities and equations. Grab your pencils, and let's get started!

1. Solving and Graphing 3x + 17 >= 5

Alright, let's kick things off with our first inequality: 3x + 17 >= 5. Our goal here is to isolate 'x' on one side of the inequality. To do this, we'll first subtract 17 from both sides:

3x + 17 - 17 >= 5 - 17

This simplifies to:

3x >= -12

Now, we'll divide both sides by 3 to solve for 'x':

3x / 3 >= -12 / 3

So we get:

x >= -4

So, our solution is x >= -4.

Graphing the Solution

To graph this inequality, we'll draw a number line. Find -4 on the number line. Since x is greater than or equal to -4, we'll use a closed circle (or a solid dot) at -4 to indicate that -4 is included in the solution. Then, we'll draw a line extending to the right from -4, indicating all values greater than -4 are also part of the solution. This line represents all the possible values of 'x' that satisfy the inequality.

Remember, the closed circle means "included," and the arrow shows the direction of all possible values. In essence, any number greater than or equal to -4 will make the original inequality true.

2. Solving and Graphing 25 - 2x < 11

Next up, we've got another inequality: 25 - 2x < 11. Again, we need to isolate 'x'. First, subtract 25 from both sides:

25 - 2x - 25 < 11 - 25

Which gives us:

-2x < -14

Now, divide both sides by -2. Remember, when we divide by a negative number in an inequality, we need to flip the inequality sign:

-2x / -2 > -14 / -2

So we get:

x > 7

Therefore, the solution is x > 7.

Graphing the Solution

To graph this, we'll use a number line again. Locate 7 on the number line. Since x is strictly greater than 7 (not equal to), we'll use an open circle at 7 to show that 7 is not included in the solution. Then, we'll draw a line extending to the right from 7, indicating that all values greater than 7 satisfy the inequality.

The open circle signifies "not included," and the arrow points towards larger values. This graph illustrates that any number greater than 7 will make the original inequality true.

3. Solving and Graphing (3/8)x < -6 or 5x > 2

Now, let's tackle a compound inequality: (3/8)x < -6 or 5x > 2. This means we have two inequalities to solve, and the solution will be any value that satisfies either one of them.

Solving (3/8)x < -6

To solve (3/8)x < -6, we'll multiply both sides by 8/3 (the reciprocal of 3/8):

(8/3) * (3/8)x < -6 * (8/3)

This simplifies to:

x < -16

Solving 5x > 2

To solve 5x > 2, we'll divide both sides by 5:

5x / 5 > 2 / 5

Which gives us:

x > 2/5

So, our combined solution is x < -16 or x > 2/5.

Graphing the Solution

On the number line, we'll have two parts to our graph. First, at -16, we'll put an open circle (since x is strictly less than -16) and draw a line extending to the left. Second, at 2/5, we'll also put an open circle (since x is strictly greater than 2/5) and draw a line extending to the right.

This graph shows two distinct regions: any number less than -16 or any number greater than 2/5 will satisfy the original compound inequality.

4. Solving and Graphing 2 < 10 - 4d < 6

Here we have another compound inequality: 2 < 10 - 4d < 6. This inequality can be solved by isolating 'd' in the middle. First, subtract 10 from all parts of the inequality:

2 - 10 < 10 - 4d - 10 < 6 - 10

This simplifies to:

-8 < -4d < -4

Now, divide all parts by -4. Remember to flip the inequality signs since we're dividing by a negative number:

-8 / -4 > -4d / -4 > -4 / -4

Which gives us:

2 > d > 1

We can rewrite this as:

1 < d < 2

So, the solution is 1 < d < 2.

Graphing the Solution

On the number line, we'll find 1 and 2. Since 'd' is strictly between 1 and 2, we'll use open circles at both 1 and 2. Then, we'll draw a line segment connecting the two circles.

This graph indicates that any number between 1 and 2 (but not including 1 or 2) will satisfy the original inequality.

5. Solving 4 - x = |2 - 3x|

Time for an equation with absolute value: 4 - x = |2 - 3x|. Remember, absolute value means the expression inside can be either positive or negative, so we need to consider two cases.

Case 1: 2 - 3x is positive or zero

In this case, |2 - 3x| = 2 - 3x, so our equation becomes:

4 - x = 2 - 3x

Add 3x to both sides:

4 - x + 3x = 2 - 3x + 3x

Which gives us:

4 + 2x = 2

Subtract 4 from both sides:

4 + 2x - 4 = 2 - 4

So we get:

2x = -2

Divide by 2:

x = -1

Case 2: 2 - 3x is negative

In this case, |2 - 3x| = -(2 - 3x) = 3x - 2, so our equation becomes:

4 - x = 3x - 2

Add x to both sides:

4 - x + x = 3x - 2 + x

Which gives us:

4 = 4x - 2

Add 2 to both sides:

4 + 2 = 4x - 2 + 2

So we get:

6 = 4x

Divide by 4:

x = 6/4 = 3/2

Now we need to check both solutions in the original equation.

For x = -1: 4 - (-1) = |2 - 3(-1)| => 5 = |5|, which is true.

For x = 3/2: 4 - (3/2) = |2 - 3(3/2)| => 5/2 = |-5/2| => 5/2 = 5/2, which is also true.

Therefore, the solutions are x = -1 and x = 3/2.

Since this equation yields specific values, there's not really a continuous graph to draw. We just mark points -1 and 3/2 on the number line to represent the solutions.

6. Solving 5|3w + 2| - 3 > 7

Last but not least, we have an inequality with absolute value: 5|3w + 2| - 3 > 7. Let's isolate the absolute value first.

Add 3 to both sides:

5|3w + 2| - 3 + 3 > 7 + 3

Which gives us:

5|3w + 2| > 10

Divide both sides by 5:

|3w + 2| > 2

Now we consider two cases:

Case 1: 3w + 2 is positive or zero

In this case, |3w + 2| = 3w + 2, so our inequality becomes:

3w + 2 > 2

Subtract 2 from both sides:

3w + 2 - 2 > 2 - 2

Which gives us:

3w > 0

Divide by 3:

w > 0

Case 2: 3w + 2 is negative

In this case, |3w + 2| = -(3w + 2), so our inequality becomes:

-(3w + 2) > 2

Multiply both sides by -1 (and flip the inequality sign):

3w + 2 < -2

Subtract 2 from both sides:

3w + 2 - 2 < -2 - 2

Which gives us:

3w < -4

Divide by 3:

w < -4/3

So, our combined solution is w > 0 or w < -4/3.

Graphing the Solution

On the number line, we'll have two parts to our graph. At 0, we'll put an open circle (since w is strictly greater than 0) and draw a line extending to the right. At -4/3, we'll also put an open circle (since w is strictly less than -4/3) and draw a line extending to the left.

This graph indicates that any number less than -4/3 or any number greater than 0 will satisfy the original inequality.

And that's a wrap, guys! We've solved and graphed a variety of inequalities and equations. Keep practicing, and you'll become math whizzes in no time! Stay tuned for more math adventures in Plastik Magazine!