Calculus Examples

$y = \sqrt{x - 1}$ , $x - y = 1$

Step 1

Solve by substitution to find the intersection between the curves.

Tap for more steps...

Step 1.1

Replace all occurrences of $y$ with $\sqrt{x - 1}$ in each equation.

Tap for more steps...

Step 1.1.1

Replace all occurrences of $y$ in $x - y = 1$ with $\sqrt{x - 1}$ .

$x - (\sqrt{x - 1}) = 1$

$y = \sqrt{x - 1}$

Step 1.1.2

Simplify the left side.

Tap for more steps...

Step 1.1.2.1

Multiply $- 1$ by $\sqrt{x - 1}$ .

$x - \sqrt{x - 1} = 1$

$y = \sqrt{x - 1}$

$x - \sqrt{x - 1} = 1$

$y = \sqrt{x - 1}$

$x - \sqrt{x - 1} = 1$

$y = \sqrt{x - 1}$

Step 1.2

Solve for $x$ in $x - \sqrt{x - 1} = 1$ .

Tap for more steps...

Step 1.2.1

Subtract $x$ from both sides of the equation.

$- \sqrt{x - 1} = 1 - x$

$y = \sqrt{x - 1}$

Step 1.2.2

To remove the radical on the left side of the equation, square both sides of the equation.

${(- \sqrt{x - 1})}^{2} = {(1 - x)}^{2}$

$y = \sqrt{x - 1}$

Step 1.2.3

Simplify each side of the equation.

Tap for more steps...

Step 1.2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 1}$ as ${(x - 1)}^{\frac{1}{2}}$ .

${(- {(x - 1)}^{\frac{1}{2}})}^{2} = {(1 - x)}^{2}$

$y = \sqrt{x - 1}$

Step 1.2.3.2

Simplify the left side.

Tap for more steps...

Step 1.2.3.2.1

Simplify ${(- {(x - 1)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 1.2.3.2.1.1

Apply the product rule to $- {(x - 1)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(x - 1)}^{\frac{1}{2}})}^{2} = {(1 - x)}^{2}$

$y = \sqrt{x - 1}$

Step 1.2.3.2.1.2

Raise $- 1$ to the power of $2$ .

$1 {({(x - 1)}^{\frac{1}{2}})}^{2} = {(1 - x)}^{2}$

$y = \sqrt{x - 1}$

Step 1.2.3.2.1.3

Multiply ${({(x - 1)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(x - 1)}^{\frac{1}{2}})}^{2} = {(1 - x)}^{2}$

$y = \sqrt{x - 1}$

Step 1.2.3.2.1.4

Multiply the exponents in ${({(x - 1)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 1.2.3.2.1.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x - 1)}^{\frac{1}{2} \cdot 2} = {(1 - x)}^{2}$

$y = \sqrt{x - 1}$

Step 1.2.3.2.1.4.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.3.2.1.4.2.1

Cancel the common factor.

${(x - 1)}^{\frac{1}{2} \cdot 2} = {(1 - x)}^{2}$

$y = \sqrt{x - 1}$

Step 1.2.3.2.1.4.2.2

Rewrite the expression.

$x - 1 = {(1 - x)}^{2}$

$y = \sqrt{x - 1}$

$x - 1 = {(1 - x)}^{2}$

$y = \sqrt{x - 1}$

$x - 1 = {(1 - x)}^{2}$

$y = \sqrt{x - 1}$

Step 1.2.3.2.1.5

Simplify.

$x - 1 = {(1 - x)}^{2}$

$y = \sqrt{x - 1}$

$x - 1 = {(1 - x)}^{2}$

$y = \sqrt{x - 1}$

$x - 1 = {(1 - x)}^{2}$

$y = \sqrt{x - 1}$

Step 1.2.3.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.3.1

Simplify ${(1 - x)}^{2}$ .

Tap for more steps...

Step 1.2.3.3.1.1

Rewrite ${(1 - x)}^{2}$ as $(1 - x) (1 - x)$ .

$x - 1 = (1 - x) (1 - x)$

$y = \sqrt{x - 1}$

Step 1.2.3.3.1.2

Expand $(1 - x) (1 - x)$ using the FOIL Method.

Tap for more steps...

Step 1.2.3.3.1.2.1

Apply the distributive property.

$x - 1 = 1 (1 - x) - x (1 - x)$

$y = \sqrt{x - 1}$

Step 1.2.3.3.1.2.2

Apply the distributive property.

$x - 1 = 1 \cdot 1 + 1 (- x) - x (1 - x)$

$y = \sqrt{x - 1}$

Step 1.2.3.3.1.2.3

Apply the distributive property.

$x - 1 = 1 \cdot 1 + 1 (- x) - x \cdot 1 - x (- x)$

$y = \sqrt{x - 1}$

$x - 1 = 1 \cdot 1 + 1 (- x) - x \cdot 1 - x (- x)$

$y = \sqrt{x - 1}$

Step 1.2.3.3.1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.2.3.3.1.3.1

Simplify each term.

Tap for more steps...

Step 1.2.3.3.1.3.1.1

Multiply $1$ by $1$ .

$x - 1 = 1 + 1 (- x) - x \cdot 1 - x (- x)$

$y = \sqrt{x - 1}$

Step 1.2.3.3.1.3.1.2

Multiply $- x$ by $1$ .

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

$y = \sqrt{x - 1}$

Step 1.2.3.3.1.3.1.3

Multiply $- 1$ by $1$ .

$x - 1 = 1 - x - x - x (- x)$

$y = \sqrt{x - 1}$

Step 1.2.3.3.1.3.1.4

Rewrite using the commutative property of multiplication.

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

$y = \sqrt{x - 1}$

Step 1.2.3.3.1.3.1.5

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.2.3.3.1.3.1.5.1

Move $x$ .

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

$y = \sqrt{x - 1}$

Step 1.2.3.3.1.3.1.5.2

Multiply $x$ by $x$ .

$x - 1 = 1 - x - x - 1 \cdot (- 1 x^{2})$

$y = \sqrt{x - 1}$

$x - 1 = 1 - x - x - 1 \cdot (- 1 x^{2})$

$y = \sqrt{x - 1}$

Step 1.2.3.3.1.3.1.6

Multiply $- 1$ by $- 1$ .

$x - 1 = 1 - x - x + 1 x^{2}$

$y = \sqrt{x - 1}$

Step 1.2.3.3.1.3.1.7

Multiply $x^{2}$ by $1$ .

$x - 1 = 1 - x - x + x^{2}$

$y = \sqrt{x - 1}$

$x - 1 = 1 - x - x + x^{2}$

$y = \sqrt{x - 1}$

Step 1.2.3.3.1.3.2

Subtract $x$ from $- x$ .

$x - 1 = 1 - 2 x + x^{2}$

$y = \sqrt{x - 1}$

$x - 1 = 1 - 2 x + x^{2}$

$y = \sqrt{x - 1}$

$x - 1 = 1 - 2 x + x^{2}$

$y = \sqrt{x - 1}$

$x - 1 = 1 - 2 x + x^{2}$

$y = \sqrt{x - 1}$

$x - 1 = 1 - 2 x + x^{2}$

$y = \sqrt{x - 1}$

Step 1.2.4

Solve for $x$ .

Tap for more steps...

Step 1.2.4.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$1 - 2 x + x^{2} = x - 1$

$y = \sqrt{x - 1}$

Step 1.2.4.2

Move all terms containing $x$ to the left side of the equation.

Tap for more steps...

Step 1.2.4.2.1

Subtract $x$ from both sides of the equation.

$1 - 2 x + x^{2} - x = - 1$

$y = \sqrt{x - 1}$

Step 1.2.4.2.2

Subtract $x$ from $- 2 x$ .

$1 + x^{2} - 3 x = - 1$

$y = \sqrt{x - 1}$

$1 + x^{2} - 3 x = - 1$

$y = \sqrt{x - 1}$

Step 1.2.4.3

Add $1$ to both sides of the equation.

$1 + x^{2} - 3 x + 1 = 0$

$y = \sqrt{x - 1}$

Step 1.2.4.4

Add $1$ and $1$ .

$x^{2} - 3 x + 2 = 0$

$y = \sqrt{x - 1}$

Step 1.2.4.5

Factor $x^{2} - 3 x + 2$ using the AC method.

Tap for more steps...

Step 1.2.4.5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $- 3$ .

$- 2, - 1$

$y = \sqrt{x - 1}$

Step 1.2.4.5.2

Write the factored form using these integers.

$(x - 2) (x - 1) = 0$

$y = \sqrt{x - 1}$

$(x - 2) (x - 1) = 0$

$y = \sqrt{x - 1}$

Step 1.2.4.6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 2 = 0$

$x - 1 = 0$

$y = \sqrt{x - 1}$

Step 1.2.4.7

Set $x - 2$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.4.7.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

$y = \sqrt{x - 1}$

Step 1.2.4.7.2

Add $2$ to both sides of the equation.

$x = 2$

$y = \sqrt{x - 1}$

$x = 2$

$y = \sqrt{x - 1}$

Step 1.2.4.8

Set $x - 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.4.8.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

$y = \sqrt{x - 1}$

Step 1.2.4.8.2

Add $1$ to both sides of the equation.

$x = 1$

$y = \sqrt{x - 1}$

$x = 1$

$y = \sqrt{x - 1}$

Step 1.2.4.9

The final solution is all the values that make $(x - 2) (x - 1) = 0$ true.

$x = 2, 1$

$y = \sqrt{x - 1}$

$x = 2, 1$

$y = \sqrt{x - 1}$

$x = 2, 1$

$y = \sqrt{x - 1}$

Step 1.3

Replace all occurrences of $x$ with $2$ in each equation.

Tap for more steps...

Step 1.3.1

Replace all occurrences of $x$ in $y = \sqrt{x - 1}$ with $2$ .

$y = \sqrt{(2) - 1}$

$x = 2$

Step 1.3.2

Simplify the right side.

Tap for more steps...

Step 1.3.2.1

Simplify $\sqrt{(2) - 1}$ .

Tap for more steps...

Step 1.3.2.1.1

Subtract $1$ from $2$ .

$y = \sqrt{1}$

$x = 2$

Step 1.3.2.1.2

Any root of $1$ is $1$ .

$y = 1$

$x = 2$

$y = 1$

$x = 2$

$y = 1$

$x = 2$

$y = 1$

$x = 2$

Step 1.4

Replace all occurrences of $x$ with $1$ in each equation.

Tap for more steps...

Step 1.4.1

Replace all occurrences of $x$ in $y = \sqrt{x - 1}$ with $1$ .

$y = \sqrt{(1) - 1}$

$x = 1$

Step 1.4.2

Simplify the right side.

Tap for more steps...

Step 1.4.2.1

Simplify $\sqrt{(1) - 1}$ .

Tap for more steps...

Step 1.4.2.1.1

Subtract $1$ from $1$ .

$y = \sqrt{0}$

$x = 1$

Step 1.4.2.1.2

Rewrite $0$ as $0^{2}$ .

$y = \sqrt{0^{2}}$

$x = 1$

Step 1.4.2.1.3

Pull terms out from under the radical, assuming positive real numbers.

$y = 0$

$x = 1$

$y = 0$

$x = 1$

$y = 0$

$x = 1$

$y = 0$

$x = 1$

Step 1.5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(2, 1)$

$(1, 0)$

$(2, 1)$

$(1, 0)$

Step 2

Solve $x - y = 1$ in terms of $y$ .

Tap for more steps...

Step 2.1

Subtract $x$ from both sides of the equation.

$- y = 1 - x$

Step 2.2

Divide each term in $- y = 1 - x$ by $- 1$ and simplify.

Tap for more steps...

Step 2.2.1

Divide each term in $- y = 1 - x$ by $- 1$ .

$\frac{- y}{- 1} = \frac{1}{- 1} + \frac{- x}{- 1}$

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{1}{- 1} + \frac{- x}{- 1}$

Step 2.2.2.2

Divide $y$ by $1$ .

$y = \frac{1}{- 1} + \frac{- x}{- 1}$

Step 2.2.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.1

Simplify each term.

Tap for more steps...

Step 2.2.3.1.1

Divide $1$ by $- 1$ .

$y = - 1 + \frac{- x}{- 1}$

Step 2.2.3.1.2

Dividing two negative values results in a positive value.

$y = - 1 + \frac{x}{1}$

Step 2.2.3.1.3

Divide $x$ by $1$ .

$y = - 1 + x$

Step 3

The area of the region between the curves is defined as the integral of the upper curve minus the integral of the lower curve over each region. The regions are determined by the intersection points of the curves. This can be done algebraically or graphically.

$A r e a = \int_{1}^{2} \sqrt{x - 1} d x - \int_{1}^{2} - 1 + x d x$

Step 4

Integrate to find the area between $1$ and $2$ .

Tap for more steps...

Step 4.1

Combine the integrals into a single integral.

$\int_{1}^{2} \sqrt{x - 1} - (- 1 + x) d x$

Step 4.2

Simplify each term.

Tap for more steps...

Step 4.2.1

Apply the distributive property.

$\sqrt{x - 1} - - 1 - x$

Step 4.2.2

Multiply $- 1$ by $- 1$ .

$\sqrt{x - 1} + 1 - x$

$\int_{1}^{2} \sqrt{x - 1} + 1 - x d x$

Step 4.3

Split the single integral into multiple integrals.

$\int_{1}^{2} \sqrt{x - 1} d x + \int_{1}^{2} d x + \int_{1}^{2} - x d x$

Step 4.4

Let $u = x - 1$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 4.4.1

Let $u = x - 1$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 4.4.1.1

Differentiate $x - 1$ .

$\frac{d}{d x} [x - 1]$

Step 4.4.1.2

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 4.4.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [- 1]$

Step 4.4.1.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$1 + 0$

Step 4.4.1.5

Add $1$ and $0$ .

$1$

Step 4.4.2

Substitute the lower limit in for $x$ in $u = x - 1$ .

$u_{lower} = 1 - 1$

Step 4.4.3

Subtract $1$ from $1$ .

$u_{lower} = 0$

Step 4.4.4

Substitute the upper limit in for $x$ in $u = x - 1$ .

$u_{upper} = 2 - 1$

Step 4.4.5

Subtract $1$ from $2$ .

$u_{upper} = 1$

Step 4.4.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = 1$

Step 4.4.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{0}^{1} \sqrt{u} d u + \int_{1}^{2} d x + \int_{1}^{2} - x d x$

Step 4.5

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

$\int_{0}^{1} u^{\frac{1}{2}} d u + \int_{1}^{2} d x + \int_{1}^{2} - x d x$

Step 4.6

By the Power Rule, the integral of $u^{\frac{1}{2}}$ with respect to $u$ is $\frac{2}{3} u^{\frac{3}{2}}$ .

$\frac{2}{3} u^{\frac{3}{2}}]_{0}^{1} + \int_{1}^{2} d x + \int_{1}^{2} - x d x$

Step 4.7

Apply the constant rule.

$\frac{2}{3} u^{\frac{3}{2}}]_{0}^{1} + x]_{1}^{2} + \int_{1}^{2} - x d x$

Step 4.8

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{2}{3} u^{\frac{3}{2}}]_{0}^{1} + x]_{1}^{2} - \int_{1}^{2} x d x$

Step 4.9

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$\frac{2}{3} u^{\frac{3}{2}}]_{0}^{1} + x]_{1}^{2} - (\frac{1}{2} x^{2}]_{1}^{2})$

Step 4.10

Combine $\frac{1}{2}$ and $x^{2}$ .

$\frac{2}{3} u^{\frac{3}{2}}]_{0}^{1} + x]_{1}^{2} - (\frac{x^{2}}{2}]_{1}^{2})$

Step 4.11

Substitute and simplify.

Tap for more steps...

Step 4.11.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $1$ and at $0$ .

$(\frac{2}{3} \cdot 1^{\frac{3}{2}}) - \frac{2}{3} \cdot 0^{\frac{3}{2}} + x]_{1}^{2} - (\frac{x^{2}}{2}]_{1}^{2})$

Step 4.11.2

Evaluate $x$ at $2$ and at $1$ .

$(\frac{2}{3} \cdot 1^{\frac{3}{2}}) - \frac{2}{3} \cdot 0^{\frac{3}{2}} + (2) - 1 - (\frac{x^{2}}{2}]_{1}^{2})$

Step 4.11.3

Evaluate $\frac{x^{2}}{2}$ at $2$ and at $1$ .

$(\frac{2}{3} \cdot 1^{\frac{3}{2}}) - \frac{2}{3} \cdot 0^{\frac{3}{2}} + 2 - 1 - (\frac{2^{2}}{2} - \frac{1^{2}}{2})$

Step 4.11.4

Simplify.

Tap for more steps...

Step 4.11.4.1

One to any power is one.

$\frac{2}{3} \cdot 1 - \frac{2}{3} \cdot 0^{\frac{3}{2}} + 2 - 1 - (\frac{2^{2}}{2} - \frac{1^{2}}{2})$

Step 4.11.4.2

Multiply $\frac{2}{3}$ by $1$ .

$\frac{2}{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}} + 2 - 1 - (\frac{2^{2}}{2} - \frac{1^{2}}{2})$

Step 4.11.4.3

Rewrite $0$ as $0^{2}$ .

$\frac{2}{3} - \frac{2}{3} \cdot {(0^{2})}^{\frac{3}{2}} + 2 - 1 - (\frac{2^{2}}{2} - \frac{1^{2}}{2})$

Step 4.11.4.4

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2}{3} - \frac{2}{3} \cdot 0^{2 (\frac{3}{2})} + 2 - 1 - (\frac{2^{2}}{2} - \frac{1^{2}}{2})$

Step 4.11.4.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.11.4.5.1

Cancel the common factor.

$\frac{2}{3} - \frac{2}{3} \cdot 0^{2 (\frac{3}{2})} + 2 - 1 - (\frac{2^{2}}{2} - \frac{1^{2}}{2})$

Step 4.11.4.5.2

Rewrite the expression.

$\frac{2}{3} - \frac{2}{3} \cdot 0^{3} + 2 - 1 - (\frac{2^{2}}{2} - \frac{1^{2}}{2})$

Step 4.11.4.6

Raising $0$ to any positive power yields $0$ .

$\frac{2}{3} - \frac{2}{3} \cdot 0 + 2 - 1 - (\frac{2^{2}}{2} - \frac{1^{2}}{2})$

Step 4.11.4.7

Multiply $0$ by $- 1$ .

$\frac{2}{3} + 0 (\frac{2}{3}) + 2 - 1 - (\frac{2^{2}}{2} - \frac{1^{2}}{2})$

Step 4.11.4.8

Multiply $0$ by $\frac{2}{3}$ .

$\frac{2}{3} + 0 + 2 - 1 - (\frac{2^{2}}{2} - \frac{1^{2}}{2})$

Step 4.11.4.9

Add $\frac{2}{3}$ and $0$ .

$\frac{2}{3} + 2 - 1 - (\frac{2^{2}}{2} - \frac{1^{2}}{2})$

Step 4.11.4.10

Subtract $1$ from $2$ .

$\frac{2}{3} + 1 - (\frac{2^{2}}{2} - \frac{1^{2}}{2})$

Step 4.11.4.11

Write $1$ as a fraction with a common denominator.

$\frac{2}{3} + \frac{3}{3} - (\frac{2^{2}}{2} - \frac{1^{2}}{2})$

Step 4.11.4.12

Combine the numerators over the common denominator.

$\frac{2 + 3}{3} - (\frac{2^{2}}{2} - \frac{1^{2}}{2})$

Step 4.11.4.13

Add $2$ and $3$ .

$\frac{5}{3} - (\frac{2^{2}}{2} - \frac{1^{2}}{2})$

Step 4.11.4.14

Raise $2$ to the power of $2$ .

$\frac{5}{3} - (\frac{4}{2} - \frac{1^{2}}{2})$

Step 4.11.4.15

Cancel the common factor of $4$ and $2$ .

Tap for more steps...

Step 4.11.4.15.1

Factor $2$ out of $4$ .

$\frac{5}{3} - (\frac{2 \cdot 2}{2} - \frac{1^{2}}{2})$

Step 4.11.4.15.2

Cancel the common factors.

Tap for more steps...

Step 4.11.4.15.2.1

Factor $2$ out of $2$ .

$\frac{5}{3} - (\frac{2 \cdot 2}{2 (1)} - \frac{1^{2}}{2})$

Step 4.11.4.15.2.2

Cancel the common factor.

$\frac{5}{3} - (\frac{2 \cdot 2}{2 \cdot 1} - \frac{1^{2}}{2})$

Step 4.11.4.15.2.3

Rewrite the expression.

$\frac{5}{3} - (\frac{2}{1} - \frac{1^{2}}{2})$

Step 4.11.4.15.2.4

Divide $2$ by $1$ .

$\frac{5}{3} - (2 - \frac{1^{2}}{2})$

Step 4.11.4.16

One to any power is one.

$\frac{5}{3} - (2 - \frac{1}{2})$

Step 4.11.4.17

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{5}{3} - (2 \cdot \frac{2}{2} - \frac{1}{2})$

Step 4.11.4.18

Combine $2$ and $\frac{2}{2}$ .

$\frac{5}{3} - (\frac{2 \cdot 2}{2} - \frac{1}{2})$

Step 4.11.4.19

Combine the numerators over the common denominator.

$\frac{5}{3} - \frac{2 \cdot 2 - 1}{2}$

Step 4.11.4.20

Simplify the numerator.

Tap for more steps...

Step 4.11.4.20.1

Multiply $2$ by $2$ .

$\frac{5}{3} - \frac{4 - 1}{2}$

Step 4.11.4.20.2

Subtract $1$ from $4$ .

$\frac{5}{3} - \frac{3}{2}$

Step 4.11.4.21

To write $\frac{5}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{5}{3} \cdot \frac{2}{2} - \frac{3}{2}$

Step 4.11.4.22

To write $- \frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{5}{3} \cdot \frac{2}{2} - \frac{3}{2} \cdot \frac{3}{3}$

Step 4.11.4.23

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 4.11.4.23.1

Multiply $\frac{5}{3}$ by $\frac{2}{2}$ .

$\frac{5 \cdot 2}{3 \cdot 2} - \frac{3}{2} \cdot \frac{3}{3}$

Step 4.11.4.23.2

Multiply $3$ by $2$ .

$\frac{5 \cdot 2}{6} - \frac{3}{2} \cdot \frac{3}{3}$

Step 4.11.4.23.3

Multiply $\frac{3}{2}$ by $\frac{3}{3}$ .

$\frac{5 \cdot 2}{6} - \frac{3 \cdot 3}{2 \cdot 3}$

Step 4.11.4.23.4

Multiply $2$ by $3$ .

$\frac{5 \cdot 2}{6} - \frac{3 \cdot 3}{6}$

Step 4.11.4.24

Combine the numerators over the common denominator.

$\frac{5 \cdot 2 - 3 \cdot 3}{6}$

Step 4.11.4.25

Simplify the numerator.

Tap for more steps...

Step 4.11.4.25.1

Multiply $5$ by $2$ .

$\frac{10 - 3 \cdot 3}{6}$

Step 4.11.4.25.2

Multiply $- 3$ by $3$ .

$\frac{10 - 9}{6}$

Step 4.11.4.25.3

Subtract $9$ from $10$ .

$\frac{1}{6}$

Step 5