Absolute Value and Inequality - Examples, Exercises and Solutions

This is an inequality problem. The inequality is actually an exercise we solve in a completely normal way, except in the case that we multiply or divide by negative.

Let's start by moving the sections:

5X+8<9

5X<9-8

5X<1

We divide by 5:

X<1/5

And this is the solution!

Inequality equations will be solved like a regular equation, except for one rule:

If we multiply the entire equation by a negative, we will reverse the inequality sign.

We start by moving the sections, so that one side has the variables and the other does not:

-3x>-10-5

-3x>-15

Divide by 3

-x>-5

Divide by negative 1 (to get rid of the negative) and remember to reverse the sign of the equation.

x<5

First, we will move the elements:

5-8x>7x+3

5-3>7x+8x
2>15x

We divide the answer by 13, and we get:

x > \frac{2}{15}

In the exercise, we have an inequality equation.

We treat the inequality as an equation with the sign -=,

And we only refer to it if we need to multiply or divide by 0.

 $10x-4 ≤ -3x-8$

We start by organizing the sections:

$10x+3x-4 ≤ -8$

$13x-4 ≤ -8$

$13x ≤ -4$

Divide by 13 to isolate the X

$x≤-\frac{4}{13}$

Let's look again at the options we were asked about:

Answer A is with different data and therefore was rejected.

Answer C shows a case where X is greater than$-\frac{4}{13}$, although we know it is small, so it is rejected.

Answer D shows a case (according to the white circle) where X is not equal to$-\frac{4}{13}$, and only smaller than it. We know it must be large and equal, so this answer is rejected.

Therefore, answer B is the correct one!

The equation is $\left|x\right|=7$, which means that the absolute value of $x$ is 7. Therefore, $x$ could be 7 or -7. Thus, the solutions are $x=7$ and $x=-7$.

The equation is $\left|x\right|=3$, which implies that $x$ can be 3 or -3. Hence, the solutions are $x=3$ and $x=-3$.

The equation is $\left|x\right|=4$. This means that the value of $x$ can be either 4 or -4. Thus, the solutions are $x=4$ and $x=-4$.

The equation is $\left|x\right|=6$, so $x$ could be 6 or -6. Therefore, the solutions are $x=6$ and $x=-6$.

To solve $\left|-3x\right|=15$, we consider both potential cases stemming from the absolute value:

1) $-3x=15$:

Divide both sides by $-3$ to get $x=-5$.

2) $-3x=-15$:

Divide both sides by $-3$ to get $x=5$.

Thus, the solutions are $x=-5$ and $x=5$.

To solve the equation $\left|-2x\right|=16$, we consider the definition of absolute value:

1. The expression inside the absolute value can be either positive or negative, but its absolute value is always positive.

2. Therefore, we set up two equations to solve:

$-2x = 16$ and $-2x = -16$

3. Solving the first equation:

$-2x = 16$

4. Divide both sides by -2:

$x = -8$

5. Solving the second equation:

$-2x = -16$

6. Divide both sides by -2:

$x = 8$

7. Therefore, the solution is:

$x=-8$ , $x=8$

To solve the equation $\left|3x\right|=21$, we consider the definition of absolute value:

1. The expression inside the absolute value can be either positive or negative, but its absolute value is always positive.

2. Therefore, we set up two equations to solve:

$3x = 21$ and $3x = -21$

3. Solving the first equation:

$3x = 21$

4. Divide both sides by 3:

$x = 7$

5. Solving the second equation:

$3x = -21$

6. Divide both sides by 3:

$x = -7$

7. Therefore, the solution is:

$x=-7$ , $x=7$

To solve the equation $\left|-4x\right|=12$, we consider the definition of absolute value:

1. The expression inside the absolute value can be either positive or negative, but its absolute value is always positive.

2. Therefore, we set up two equations to solve:

$-4x = 12$ and $-4x = -12$

3. Solving the first equation:

$-4x = 12$

4. Divide both sides by -4:

$x = -3$

5. Solving the second equation:

$-4x = -12$

6. Divide both sides by -4:

$x = 3$

7. Therefore, the solution is:

$x=-3$ , $x=3$

Let's solve the equation $-\left|x\right|=15$. Since the expression inside the absolute value can be either positive or negative, we consider two scenarios:

1. $-x = 15$: This implies $x = -15$.

2. $-(-x) = 15$: This simplifies to $x = 15$.

Thus, the solutions are $x = -15$ and $x = 15$.

Let's solve the equation $-\left|x\right|=8$. The expression inside the absolute value can take two forms:

1. $-x = 8$: This gives $x = -8$.

2. $-(-x) = 8$: This simplifies to $x = 8$.

Therefore, the solutions are $x = -8$ and $x = 8$.

Solving an inequality equation is just like a normal equation. We start by trying to isolate the variable (X).

It is important to note that in this equation there are two variables (X and a), so we may not reach a final result.

 8x+a<3x-4

We move the sections

8x-3x<-4-a

We reduce the terms

5x<-4-a

We divide by 5

x< -a/5 -4/5

And this is the solution!