Solved on Jan 25, 2024

Solve the absolute value equation $|h-4|=7$ . Rewrite as a compound statement, solve for $h$ , and write the solution set.

STEP 1

Assumptions
1. We are given the absolute value equation $|h-4|=7$ .
2. We need to rewrite the equation as a compound statement.
3. We need to solve the resulting linear equations for $h$ .
4. We need to express the solution set in set-builder notation.

STEP 2

The definition of absolute value states that $|x|=a$ has two solutions: $x=a$ or $x=-a$ if $a \geq 0$ . We can apply this definition to the given absolute value equation.

STEP 3

Rewrite the absolute value equation as a compound statement, which means we will have two separate equations without the absolute value.
$h - 4 = 7 \quad \text{or} \quad h - 4 = -7$

STEP 4

Now, we solve the first linear equation for $h$ .
$h - 4 = 7$

STEP 5

Add 4 to both sides of the equation to isolate $h$ .
$h = 7 + 4$

STEP 6

Calculate the value of $h$ for the first equation.
$h = 11$

STEP 7

Next, we solve the second linear equation for $h$ .
$h - 4 = -7$

STEP 8

Add 4 to both sides of the equation to isolate $h$ .
$h = -7 + 4$

STEP 9

Calculate the value of $h$ for the second equation.
$h = -3$

STEP 10

Now that we have the two solutions, we can express the solution set in set-builder notation.

STEP 11

Write the solution set in set-builder notation, which includes all the elements that satisfy the original equation.
$\{h \mid h = 11 \text{ or } h = -3\}$
The solution set in set-builder notation is $\{h \mid h = 11 \text{ or } h = -3\}$ .

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Solve the absolute value equation $|h-4|=7$ . Rewrite as a compound statement, solve for $h$ , and write the solution set.

STEP 1

Assumptions
1. We are given the absolute value equation $|h-4|=7$ .
2. We need to rewrite the equation as a compound statement.
3. We need to solve the resulting linear equations for $h$ .
4. We need to express the solution set in set-builder notation.

STEP 2

The definition of absolute value states that $|x|=a$ has two solutions: $x=a$ or $x=-a$ if $a \geq 0$ . We can apply this definition to the given absolute value equation.

STEP 3

Rewrite the absolute value equation as a compound statement, which means we will have two separate equations without the absolute value.
$h - 4 = 7 \quad \text{or} \quad h - 4 = -7$

STEP 4

Now, we solve the first linear equation for $h$ .
$h - 4 = 7$

STEP 5

Add 4 to both sides of the equation to isolate $h$ .
$h = 7 + 4$

STEP 6

Calculate the value of $h$ for the first equation.
$h = 11$

STEP 7

Next, we solve the second linear equation for $h$ .
$h - 4 = -7$

STEP 8

Add 4 to both sides of the equation to isolate $h$ .
$h = -7 + 4$

STEP 9

Calculate the value of $h$ for the second equation.
$h = -3$

STEP 10

Now that we have the two solutions, we can express the solution set in set-builder notation.

STEP 11

Write the solution set in set-builder notation, which includes all the elements that satisfy the original equation.
$\{h \mid h = 11 \text{ or } h = -3\}$
The solution set in set-builder notation is $\{h \mid h = 11 \text{ or } h = -3\}$ .

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Solve the absolute value equation ∣h−4∣=7|h-4|=7∣h−4∣=7. Rewrite as a compound statement, solve for hhh, and write the solution set.

STEP 1

Assumptions1. We are given the absolute value equation ∣h−4∣=7|h-4|=7∣h−4∣=7.2. We need to rewrite the equation as a compound statement.3. We need to solve the resulting linear equations for hhh.4. We need to express the solution set in set-builder notation.

STEP 2

The definition of absolute value states that ∣x∣=a|x|=a∣x∣=a has two solutions: x=ax=ax=a or x=−ax=-ax=−a if a≥0a \geq 0a≥0. We can apply this definition to the given absolute value equation.

STEP 3

Rewrite the absolute value equation as a compound statement, which means we will have two separate equations without the absolute value.h−4=7orh−4=−7h - 4 = 7 \quad \text{or} \quad h - 4 = -7h−4=7orh−4=−7

STEP 4

Now, we solve the first linear equation for hhh.h−4=7h - 4 = 7h−4=7

STEP 5

Add 4 to both sides of the equation to isolate hhh.h=7+4h = 7 + 4h=7+4

STEP 6

Calculate the value of hhh for the first equation.h=11h = 11h=11

STEP 7

Next, we solve the second linear equation for hhh.h−4=−7h - 4 = -7h−4=−7

STEP 8

Add 4 to both sides of the equation to isolate hhh.h=−7+4h = -7 + 4h=−7+4

STEP 9

Calculate the value of hhh for the second equation.h=−3h = -3h=−3

STEP 10

Now that we have the two solutions, we can express the solution set in set-builder notation.

STEP 11

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Solve the absolute value equation $|h-4|=7$ . Rewrite as a compound statement, solve for $h$ , and write the solution set.

Assumptions
1. We are given the absolute value equation $|h-4|=7$ .
2. We need to rewrite the equation as a compound statement.
3. We need to solve the resulting linear equations for $h$ .
4. We need to express the solution set in set-builder notation.

The definition of absolute value states that $|x|=a$ has two solutions: $x=a$ or $x=-a$ if $a \geq 0$ . We can apply this definition to the given absolute value equation.

Rewrite the absolute value equation as a compound statement, which means we will have two separate equations without the absolute value.
$h - 4 = 7 \quad \text{or} \quad h - 4 = -7$

Now, we solve the first linear equation for $h$ .
$h - 4 = 7$

Add 4 to both sides of the equation to isolate $h$ .
$h = 7 + 4$

Calculate the value of $h$ for the first equation.
$h = 11$

Next, we solve the second linear equation for $h$ .
$h - 4 = -7$

Add 4 to both sides of the equation to isolate $h$ .
$h = -7 + 4$

Calculate the value of $h$ for the second equation.
$h = -3$