Question

1. $-|m + 3|=-13$
2. $|-4m|=64$
3. $2|x + 1|-7=-3$
4. $5|c + 3|-1=9$
5. $-2|2p - 3|-1=-11$
6. explain why the equation $|m|=-3$ has no solution.

Explanation:

Problem 7: Isolate absolute value

Multiply both sides by -1:
$|m+3|=13$

Problem 7: Solve absolute value

Split into two equations:
$m+3=13$ or $m+3=-13$
Solve: $m=13-3=10$ or $m=-13-3=-16$

Problem 8: Simplify absolute value

Use $|-a|=|a|$:
$|4m|=64$

Problem 8: Solve for m

Divide by 4: $|m|=16$
Split into two equations:
$m=16$ or $m=-16$

Problem 9: Isolate absolute value

Add 7 to both sides:
$2|x+1|=4$
Divide by 2: $|x+1|=2$

Problem 9: Solve absolute value

Split into two equations:
$x+1=2$ or $x+1=-2$
Solve: $x=2-1=1$ or $x=-2-1=-3$

Problem 10: Isolate absolute value

Add 1 to both sides:
$5|c+3|=10$
Divide by 5: $|c+3|=2$

Problem 10: Solve absolute value

Split into two equations:
$c+3=2$ or $c+3=-2$
Solve: $c=2-3=-1$ or $c=-2-3=-5$

Problem 11: Isolate absolute value

Add 1 to both sides:
$-2|2p-3|=-10$
Divide by -2: $|2p-3|=5$

Problem 11: Solve absolute value

Split into two equations:
$2p-3=5$ or $2p-3=-5$
Solve: $2p=8 \implies p=4$ or $2p=-2 \implies p=-1$

Problem 12: Define absolute value

Absolute value of a number is its non-negative distance from 0 on the number line.

Problem 12: Contradict given equation

The equation $|m|=-3$ claims a non-negative value equals a negative number, which is impossible.

Answer:

1. $m=10$ or $m=-16$
2. $m=16$ or $m=-16$
3. $x=1$ or $x=-3$
4. $c=-1$ or $c=-5$
5. $p=4$ or $p=-1$
6. The absolute value of any real number is always non-negative (greater than or equal to 0), so it can never equal a negative number like -3, hence no solution exists.