Absolute Value Inequalities Practice Problems & Solutions

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 this inequality, we use the principle that if $|A| > B$, then $A > B$ or $A < -B$. Here's a detailed step-by-step solution:

• Set up the two inequalities from the absolute value expression:
  1. $4x + 12 > 16$
  2. $4x + 12 < -16$
• For the first inequality, $4x + 12 > 16$:
  - Subtract 12 from both sides to isolate the term with $x$:
  $4x > 16 - 12$
  $4x > 4$
  - Divide both sides by 4:
  $x > 1$
• For the second inequality, $4x + 12 < -16$:
  - Subtract 12 from both sides:
  $4x < -16 - 12$
  $4x < -28$
  - Divide both sides by 4:
  $x < -7$
• The solution is the union of both results:
  $x > 1$ or $x < -7$

Therefore, the correct answer among the given choices is $x > 1$ or $x < -7$.

To solve the inequality $\left|5x-10\right| > 15$, we rewrite it as two separate inequalities:

• First inequality: $5x - 10 > 15$
• Second inequality: $5x - 10 < -15$

Let's solve each one:

For the first inequality $5x - 10 > 15$:
Add 10 to both sides: $5x > 25$
Divide both sides by 5: $x > 5$

For the second inequality $5x - 10 < -15$:
Add 10 to both sides: $5x < -5$
Divide both sides by 5: $x < -1$

Combining these solutions, we have:
$x > 5$ or $x < -1$

Therefore, the correct statement regarding the solution set is: $x > 5$ or $x < -1$.

To solve the inequality $\left|3x + 9\right| < 18$, follow these steps:

• Step 1: Remove the absolute value by expressing it as a double inequality:
  $-18 < 3x + 9 < 18$.

• Step 2: Simplify the inequality:
  First, subtract 9 from all parts:
  $-18 - 9 < 3x + 9 - 9 < 18 - 9$,
  which simplifies to $-27 < 3x < 9$.

• Step 3: Solve for $x$ by dividing the entire inequality by 3:
  $-\frac{27}{3} < \frac{3x}{3} < \frac{9}{3}$,
  resulting in $-9 < x < 3$.

Upon solving, we determine that the solution to the inequality is the interval:

$-9 < x < 3$.

To solve the absolute value inequality $|2x - 4| < 8$, we begin by removing the absolute value expression. This gives us a compound inequality:

$-8 < 2x - 4 < 8$.

We will solve this compound inequality by handling each part separately:

• Start with the left inequality: $-8 < 2x - 4$.
• Add 4 to both sides to isolate the term with $x$: $-8 + 4 < 2x$.
• Simplify: $-4 < 2x$.
• Finally, divide both sides by 2: $-2 < x$.

• Now, solve the right inequality: $2x - 4 < 8$.
• Add 4 to both sides to isolate the term with $x$: $2x - 4 + 4 < 8 + 4$.
• Simplify: $2x < 12$.
• Finally, divide both sides by 2: $x < 6$.

Combining the two solutions from the parts, we find:

$-2 < x < 6$.

The solution indicates that $x$ must be greater than -2 and less than 6. This form matches answer choice 4. Therefore, the correct solution is:

$-2 < x < 6$.