Equations with Absolute Values: Solving the problem

To solve the equation $\left| x \right| = 5$, consider what absolute value means. The absolute value $\left| x \right|$ represents the distance between $x$ and 0 on the number line, meaning it’s always non-negative.

When solving $\left| x \right| = 5$, we find the values of $x$ that are 5 units away from 0. Hence, the absolute value equation $\left| x \right| = 5$ results in two possible equations:

• $x = 5$
• $x = -5$

So, the solutions to the absolute value equation $\left| x \right| = 5$ are $x = 5$ and $x = -5$.

Let's compare these solutions to the answer choices provided:

• Choice 1: $x = 5$ matches one solution.
• Choice 2: $x = -5$ matches the other solution.
• Choice 4: "Answers a + b" suggests both are correct, which is indeed the case.

Thus, the choice that correctly represents the solutions to the equation $\left| x \right| = 5$ is "Answers a + b".

Therefore, the correct answer is:

Answers a + b

To solve this problem, we'll follow these steps:

• Step 1: Use the definition of absolute value to create equations.
• Step 2: Solve each equation to find possible values of $x$.
• Step 3: Check the solutions against the original problem.

Now, let's work through each step:
Step 1: The equation is given as $\left| -x \right| = 10$. According to the definition of absolute value:

• Case 1: $-x = 10$
• Case 2: $-x = -10$

Step 2: Solve each case:
In Case 1, we have:

• $-x = 10$
  This implies $x = -10$.

In Case 2, we have:

• $-x = -10$
  This implies $x = 10$.

Step 3: Therefore, the possible solutions are $x = -10$ and $x = 10$, which satisfy the equation independently.

The solution to the problem is:

$x = -10$ , $x = 10$

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

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 $x=-4$.

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 $x = 8$.

To solve the absolute value equation $|x + 1| = 5$, we follow these steps:

• Step 1: Understand that the equation $|x + 1| = 5$ means the expression inside the absolute value, $x + 1$, is equal to 5 or -5.
• Step 2: Set up two separate equations:
• Equation 1: $x + 1 = 5$
• Equation 2: $x + 1 = -5$
• Step 3: Solve each equation individually.
• For Equation 1: Subtract 1 from both sides to get $x = 5 - 1 = 4$.
• For Equation 2: Subtract 1 from both sides to get $x = -5 - 1 = -6$.

Thus, the solutions to the equation $|x + 1| = 5$ are $x = 4$ and $x = -6$.

Therefore, the correct answer, considering the choices provided, is Answer a + b, which corresponds to choices 1 and 2.

To solve the problem $\left|x-10\right|=0$, we use this logical reasoning:

• Step 1: Recognize that the absolute value expression equals zero.
  The property of absolute values states that for $\left|A\right|=0$, $A$ must equal zero.
• Step 2: Apply this principle to the given equation.
  Given $\left|x-10\right|=0$, it follows that $x-10=0$.
• Step 3: Solve for $x$.
  Rearrange the equation $x - 10 = 0$ to find $x = 10$.

The solution to the equation $\left|x-10\right|=0$ is therefore $x = 10$.

To solve the equation $|x - 3| = 4$, follow these steps:

• Step 1: Understand that $|x - 3| = 4$ means $x - 3$ can be either 4 or -4.
• Step 2: Set up two separate equations:
• Equation 1: $x - 3 = 4$
• Equation 2: $x - 3 = -4$
• Step 3: Solve each equation for $x$.

Let's solve Equation 1:
$x - 3 = 4$
Add 3 to both sides:
$x = 4 + 3$
$x = 7$

Now, solve Equation 2:
$x - 3 = -4$
Add 3 to both sides:
$x = -4 + 3$
$x = -1$

Therefore, the solutions to the equation $|x - 3| = 4$ are $x = 7$ and $x = -1$.

Checking the given choices, the correct answer is:
$x = -1$, $x = 7$, which matches choice 1.

To solve this problem, we'll follow these steps:

• Step 1: Set up two linear equations from the absolute value definition.
• Step 2: Solve each equation for $x$.

Now, let's work through each step:

Step 1: Consider the two cases from $\left|x + 2\right| = 4$.

• First case: $x + 2 = 4$.
• Second case: $x + 2 = -4$.

Step 2: Solve the two equations:

For the first case:

Subtract 2 from both sides:

For the second case:

Subtract 2 from both sides:

Therefore, the solutions to the equation $\left|x + 2\right| = 4$ are:

$x = -6$ and $x = 2$.

The correct option is:

$x = -6, x = 2$

Thus, the solution to the problem is:

$x = -6, x = 2$

To solve $\left| z + 4 \right| = 12$, consider the two cases:

1. $z + 4 = 12$ gives $z = 8$

2. $z + 4 = -12$ gives $z = -16$

Thus, the solutions are $z = 8$ and $z = -16$.

To solve $\left| y - 3 \right| = 7$, we need to consider the two possible cases for the absolute value equation.

1. $y - 3 = 7$ leads to $y = 10$

2. $y - 3 = -7$ leads to $y = -4$

Thus, the solutions are $y = 10$ and $y = -4$.

To solve $\left| a - 2 \right| = 6$, consider the cases:

1. $a - 2 = 6$ leads to $a = 8$

2. $a - 2 = -6$ leads to $a = -4$

Thus, the solutions are $a = 8$ and $a = -4$.

To solve this problem, we'll use the absolute value property, which splits the equation into two separate cases:

• Case 1: Solve $5 - x = 7$

Subtract 5 from both sides:

$5 - x - 5 = 7 - 5$

Which simplifies to:

$-x = 2$

Multiplying both sides by -1 gives:

$x = -2$

• Case 2: Solve $5 - x = -7$

Subtract 5 from both sides:

$5 - x - 5 = -7 - 5$

Which simplifies to:

$-x = -12$

Multiplying both sides by -1 gives:

$x = 12$

Thus, the solutions are $x = -2$ and $x = 12$. Now let's verify against the available choices. We can see that:

1. x = 0
2. x = -2
3. x = 12
4. Answers b + c (i.e., x = -2 and x = 12)

This confirms that the correct choice is "Answers b + c".

Therefore, the correct answer is Answers b + c.