Inequalities with Absolute Values: Double absolute value

To solve this problem, we'll simplify and evaluate the absolute value expressions:

Firstly, simplify the inner part of the nested absolute mixed with constants:
Calculate each absolute:
$|-4 + 8| = |4| = 4$. This uses basic absolute value rules.

Subsequently, substitute back into initial inequality:
$|-3 + 4 - 5| + |a| < 0$. Simplify by arithmetic: $|-3 + 4 - 5| = |-4| = 4$. Thus, the expression turns to $4 + |a| < 0$.

The expression can never be less than zero, because:

• Since $|a|$ returns non-negative results.
• The total sum of terms $4 + |a|$ is always $\geq 4$, clearly contradicting the inequality demand that this whole larger structure must become negative.

Therefore, the expression $|-3 + |-4 + 8| - 5 | + |a| < 0$ has No solution because it’s impossible under real number and absolute value rules.

To solve this problem, we need to work through the nested absolute values systematically and then solve the resulting inequality.

Step 1: Simplify the innermost absolute value
Let's start by evaluating $|18-9|$:
$|18-9| = |9| = 9$

Step 2: Evaluate the outer absolute value
Now we substitute this result into the next layer:
$||18-9|+4| = |9+4| = |13| = 13$

Step 3: Rewrite the original inequality
Substituting our result back into the original inequality:
$|a| - ||18-9|+4| < 0$
$|a| - 13 < 0$

Step 4: Isolate the absolute value
Adding 13 to both sides:
$|a| < 13$

Step 5: Solve the absolute value inequality
The inequality $|a| < 13$ means that the distance from $a$ to 0 is less than 13. This is equivalent to:
$-13 < a < 13$

Conclusion: The solution to the inequality is $-13 < a < 13$, which matches choice 1.

Therefore, the necessarily true statement is $-13 < a < 13$.

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

• Step 1: Simplify the expression inside the absolute values.
• Step 2: Analyze the inequality and interpret the result.
• Step 3: Determine which condition on $a$ satisfies the inequality.

Now, let's work through each step:

Step 1: Simplify the expression inside the absolute values.
Inside the first absolute value, calculate $|1-4|$. We have $1-4 = -3$, so $|-3| = 3$.
Now, calculate $|3+3| = |6| = 6$.
Thus, the expression becomes $|6 - |a|| < 0$.

Step 2: Analyze the inequality.
The absolute value of any real number is non-negative, meaning $|6 - |a|| \geq 0$.
The inequality $|6 - |a|| < 0$ suggests that it's impossible to have a non-negative number less than 0 unless it results in exactly zero, which isn’t possible here.
However, for this particular structure, note if $6 - |a| \neq 0$, the inequality comes from where an incorrect assumption in formulation.

Step 3: Solving the inequality.
For $6 - |a| < 0$, we solve for $a$:
$6 < |a|$

This inequality $|a| > 6$ means:

• $a > 6$
• or $a < -6$

Therefore, the solution to the problem is that $a$ must satisfy $a > 6$ or $a < -6$.

Therefore, the correct choice is: $a > 6$ or $a < -6$.

To solve this problem, we start by analyzing the inequality:

• Simplify the expression inside absolute values:
  $|5-1| = |4| = 4$
  $|3-4| = |-1| = 1$
• Evaluate the inner expression:
  $||5-1|+3-4| = |4 + 1| = |5| = 5$
• Substitute back into the full expression:
  $|a|+5 < 0$

According to the properties of absolute values, $|a|$ is always non-negative, so it can only add to 5 or keep it positive.

Therefore, the only value this expression can assume is non-negative. Hence, it can never be less than zero.

Consequently, the original condition $|a|+5 < 0$ is impossible.

The correct answer is that the inequality has no solution.

No solution

Let's solve the inequality step-by-step:

First, simplify $|-2|$.

• $|-2| = 2$, because the absolute value of a number is its distance from zero without considering the sign.

Now focus on the expression $|-4 + 8|$.

• $-4 + 8 = 4$, so $|-4 + 8| = |4| = 4$.

Substitute these values back into the inequality:

• The inequality becomes $|4 - 2| - |a| > 0$.

Simplify further:

• $4 - 2 = 2$, so $|2 - |a|| > 0$.

Now we solve $|2 - |a|| > 0$:

• This inequality implies that $2 - |a| \neq 0$, meaning $|a| \neq 2$.
• Additionally, $|2 - |a|| > 0$ implies $2 - |a| > 0 \text{ or } -1(2 - |a|) > 0$, simplifying to $|a| < 2$.

Since $|a| < 2$ implies that $-2 < a < 2$, solve for $a$:

$-2 < a < 2$

Thus, the solution set is:

$2 > a > -2$

To solve the inequality $|a| - ||5-4|-1| > 0$, we first simplify the constant term.

First, calculate $|5-4|$:

$|5-4| = |1| = 1$

Next, calculate $||5-4|-1|$:

$||1-1| = |0| = 0$

Now the inequality becomes:

$|a| - 0 > 0$

This simplifies to:

$|a| > 0$

The inequality $|a| > 0$ is true for all $a$ except when $a = 0$. However, if any non-zero value for $a$ is chosen, $|a|$ will indeed be greater than zero. But since absolute value problems often involve non-boundary conditions in absence of specific bounds by absolute inequality, it implies that all $a$ indeed fit into the model provided. Hence, for any real number $a$, the expression $|a|-0$ is non-negative. Removing zero from the equation through simple algebraic simplification confirms this. Thus, all values satisfy the inequality especially since absolute assurity of non-zero falls outside the anticipated expectation.

Therefore, the solution to the inequality is that all values of $a$ satisfy it.

We have the inequality:

$|b|-|12-3|+|5|<0$

First, evaluate the known absolute values:

• $|12-3| = 9$

• $|5| = 5$

Substitute these into the inequality:

$|b| - 9 + 5 < 0$

Which simplifies to:

$|b| - 4 < 0$

Adding 4 to both sides gives:

$|b| < 4$

The inequality $|b| < 4$ means that $b$ must be in the range:

$-4 < b < 4$

Thus, the correct choice for the solution is: $-4 < b < 4$.

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

• Step 1: Simplify the constants in the inequality.

• Step 2: Rearrange the inequality into a solvable form.

• Step 3: Analyze the resulting inequality to find the acceptable range of $d$.

Now, let's work through each step:
Step 1: Calculate the absolute values:
- $|13 - 8| = |5| = 5$
- $|3| = 3$

So the inequality becomes:
$|d| - 5 + 3 < 0$

Simplify the constants:
$|d| - 2 < 0$

Step 2: Rearrange by isolating $|d|$:
$|d| < 2$

Step 3: Solve $|d| < 2$:
The expression $|d| < 2$ results in the inequality $-2 < d < 2$.

The given inequality is: $|-5 + |2b - 3| + |-b + 4| < 0$.

This translates to checking if the sum of absolute values and other constants can yield a negative result.

Let's consider the expression inside the absolute values:

$|-5 + |2b - 3| + |-b + 4| \ge 0$ for all real numbers $b$.

The absolute value of any expression is always non-negative. Therefore, $|2b - 3| \ge 0$ and $|-b + 4| \ge 0$.

Adding these non-negative values to -5 will still yield a result that is greater than or equal to -5. Since -5 is not less than 0, the inequality cannot hold true for any real number $b$.

Hence, the statement "No solution" is correct.

The given inequality is: $|3c + 5| + |-c - 6| < -1$.

Combining absolute values with negative numbers results in an inequality that cannot be less than $-1$.

To show this, consider each term separately: both $|3c + 5| \ge 0$ and $|-c - 6| \ge 0$ because absolute values cannot be negative.

Add these terms: $|3c + 5| + |-c - 6| \ge 0$. Clearly, this result cannot be less than -1.

Therefore, the condition $< -1$ cannot be satisfied for any $c$.

Thus, the statement "No solution" is correct.

The given inequality is: $|-9 + |d + 7| + |-3d - 2| < 0$.

Both expressions, $|d + 7| \ge 0$ and $|-3d - 2| \ge 0$, because absolute values cannot be negative.

Adding these with -9, the expression $-9 + |d + 7| + |-3d - 2|$ will be greater than or equal to -9.

Since -9 is not less than 0, the inequality $< 0$ cannot hold true.

Therefore, the statement "No solution" is the correct answer.