Inequalities with Absolute Values: Double absolute value

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.

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'll follow these steps:

• Simplify the expression inside the inequality.
• Determine the necessary condition for $a$.
• Compare the result with the provided options.

Step 1: Simplify the expression.
We start by evaluating the fixed absolute values:
$|18 - 9| = 9$ and $|4| = 4$.
Substituting these values into the inequality gives us:
$|a| - 9 + 4 < 0.$

Step 2: Simplify further and solve for $|a|$.
Combine constants:
$|a| - 5 < 0$
Thus, we have:
$|a| < 5$.

Step 3: Apply the property of absolute values.
The inequality $|a| < 5$ implies that:
$-5 < a < 5$.

Therefore, the solution to the problem is $-5 < a < 5$.

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.

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

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.