Pre-Algebra Examples

$h (x) = (3 x - 17) - (- 14 + 6 x)$

Step 1

Check the leading coefficient of the function. This number is the coefficient of the expression with the largest degree.

Largest Degree: $1$

Leading Coefficient: $1$

Step 2

Check the leading coefficient of the function. This number is the coefficient of the expression with the largest degree.

Largest Degree: $1$

Leading Coefficient: $- 1$

Step 3

The leading coefficient needs to be $1$ . If it is not, divide the expression by it to make it $1$ .

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Step 3.1

Combine the numerators over the common denominator.

$h (x) = \frac{3 x - 17 - (- 14 + 6 x)}{- 1}$

Step 3.2

Simplify each term.

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Step 3.2.1

Apply the distributive property.

$h (x) = \frac{3 x - 17 + 14 - (6 x)}{- 1}$

Step 3.2.2

Multiply $- 1$ by $- 14$ .

$h (x) = \frac{3 x - 17 + 14 - (6 x)}{- 1}$

Step 3.2.3

Multiply $6$ by $- 1$ .

$h (x) = \frac{3 x - 17 + 14 - 6 x}{- 1}$

Step 3.3

Subtract $6 x$ from $3 x$ .

$h (x) = \frac{- 3 x - 17 + 14}{- 1}$

Step 3.4

Add $- 17$ and $14$ .

$h (x) = \frac{- 3 x - 3}{- 1}$

Step 3.5

Move the negative one from the denominator of $\frac{- 3 x - 3}{- 1}$ .

$h (x) = - 1 \cdot (- 3 x - 3)$

Step 3.6

Rewrite $- 1 \cdot (- 3 x - 3)$ as $- (- 3 x - 3)$ .

$h (x) = - (- 3 x - 3)$

Step 3.7

Apply the distributive property.

$h (x) = - (- 3 x) + 3$

Step 3.8

Multiply $- 3$ by $- 1$ .

$h (x) = 3 x + 3$

Step 3.9

Multiply $- 1$ by $- 3$ .

$h (x) = 3 x + 3$

Step 4

Create a list of the coefficients of the function except the leading coefficient of $1$ .

$3$

Step 5

There will be two bound options, $b_{1}$ and $b_{2}$ , the smaller of which is the answer. To calculate the first bound option, find the absolute value of the largest coefficient from the list of coefficients. Then add $1$ .

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Step 5.1

Arrange the terms in ascending order.

$b_{1} = | 3 |$

Step 5.2

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$b_{1} = 3 + 1$

Step 5.3

Add $3$ and $1$ .

$b_{1} = 4$

Step 6

To calculate the second bound option, sum the absolute values of the coefficients from the list of coefficients. If the sum is greater than $1$ , use that number. If not, use $1$ .

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Step 6.1

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$b_{2} = 3$

Step 6.2

Arrange the terms in ascending order.

$b_{2} = 1, 3$

Step 6.3

The maximum value is the largest value in the arranged data set.

$b_{2} = 3$

Step 7

Take the smaller bound option between $b_{1} = 4$ and $b_{2} = 3$ .

Smaller Bound: $3$

Step 8

Every real root on $h (x) = (3 x - 17) - (- 14 + 6 x)$ lies between $- 3$ and $3$ .

$- 3$ and $3$