Absolute Value and Inequality Practice Problems with Solutions

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

Let's solve the problem:
Step 1: Recognize that the inequality we are dealing with is $\left|2x-1\right| > -10$.
Step 2: Consider the nature of absolute values: for any real $x$, $\left|2x-1\right|$ is always non-negative (i.e., $\geq 0$).
Step 3: Observe that the right side of the inequality, $-10$, is negative. Therefore, the inequality $\left|2x-1\right| > -10$ is always true because $\left|2x-1\right|$ as a non-negative quantity will always be greater than any negative number.
Step 4: Since the inequality condition always holds true, this means that the statement is valid for all $x$.

Therefore, the correct answer is that the inequality holds for all $x$.

The solution to the problem is For all x.

To solve the inequality $|x + 3| \leq 5$, we first consider the definition of absolute value inequality $|A| \leq B$, which is equivalent to $-B \leq A \leq B$.

Applying this definition, we have:

$-5 \leq x + 3 \leq 5$

Next, we isolate x by subtracting 3 from all parts of the inequality:

$-5 - 3 \leq x + 3 - 3 \leq 5 - 3$

This simplifies to:

$-8 \leq x \leq 2$

To solve the inequality $|x - 4| < 8$, we will break it down into two separate inequalities.

• First, recognize that $|x - 4| < 8$ means the expression $x - 4$ can vary between -8 and 8 without violating the inequality constraint.
• This gives us two inequalities to solve: $-8 < x - 4$ and $x - 4 < 8$.

Let's solve each inequality:
1. For $-8 < x - 4$:
- Add 4 to both sides to isolate $x$:
$-8 + 4 < x \rightarrow -4 < x$ 2. For $x - 4 < 8$:
- Add 4 to both sides to isolate $x$:
$x < 8 + 4 \rightarrow x < 12$

By combining these results, we obtain the solution:
$-4 < x < 12$

Therefore, the range of $x$ that satisfies the inequality $|x - 4| < 8$ is $-4 < x < 12$.

Hence, the correct statement from the given choices is $\boxed{-4 < x < 12}$.

To solve the inequality $|x-3| \leq 5$, we need to consider the definition of absolute value inequalities. The inequality $|a| \leq b$ translates to $-b \leq a \leq b$.

Applying this to our expression $|x-3| \leq 5$, we have:

$-5 \leq x-3 \leq 5$.

We add 3 to all parts of the inequality to isolate $x$:

$-5 + 3 \leq x - 3 + 3 \leq 5 + 3$

This simplifies to $-2 \leq x \leq 8$.