Algebra Examples

$\sqrt{x + y} \leq 1$

Step 1

Solve for $y$ .

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Step 1.1

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

${\sqrt{x + y}}^{2} \leq 1^{2}$

Step 1.2

Simplify each side of the inequality.

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Step 1.2.1

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

${({(x + y)}^{\frac{1}{2}})}^{2} \leq 1^{2}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

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

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Step 1.2.2.1.1

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

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Step 1.2.2.1.1.1

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

${(x + y)}^{\frac{1}{2} \cdot 2} \leq 1^{2}$

Step 1.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 1.2.2.1.1.2.1

Cancel the common factor.

${(x + y)}^{\frac{1}{2} \cdot 2} \leq 1^{2}$

Step 1.2.2.1.1.2.2

Rewrite the expression.

${(x + y)}^{1} \leq 1^{2}$

Step 1.2.2.1.2

Simplify.

$x + y \leq 1^{2}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

One to any power is one.

$x + y \leq 1$

Step 1.3

Subtract $x$ from both sides of the inequality.

$y \leq 1 - x$

Step 2

Find the slope and the y-intercept for the boundary line.

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Step 2.1

Rewrite in slope-intercept form.

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Step 2.1.1

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Step 2.1.2

Reorder $1$ and $- x$ .

$y \leq - x + 1$

Step 2.2

Use the slope-intercept form to find the slope and y-intercept.

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Step 2.2.1

Find the values of $m$ and $b$ using the form $y = m x + b$ .

$m = - 1$

$b = 1$

Step 2.2.2

The slope of the line is the value of $m$ , and the y-intercept is the value of $b$ .

Slope: $- 1$

y-intercept: $(0, 1)$

Slope: $- 1$

y-intercept: $(0, 1)$

Slope: $- 1$

y-intercept: $(0, 1)$

Step 3

Graph a solid line, then shade the area below the boundary line since $y$ is less than $1 - x$ .

$y \leq 1 - x$

Step 4