Question

Which linear inequality is represented by the graph? (A)y ≤1/3 x − 1 (B)y ≥ x − 1 (C)y < 3x − 1 (D)y > 3x − 1

Answer 1

The equation of the line is $\boxed{y \geqslant \frac{1}{3}x - 1}.$

Further explanation:

The linear equation with slope $m$ and intercept $c$ is given as follows.

$\boxed{y = mx + c}$

The formula for slope of line with points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ can be expressed as,

$\boxed{m = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}}$

Given:

Explanation:

The line intersects y-axis at $\left( {0, - 1} \right)$. Therefore, the y-intercept is $-1.$

The line passes through the points $\left( {0, - 1} \right)$ and $\left( {3,0} \right).$

The slope can be obtained as follows,

$\begin{aligned}m&= \frac{{0- \left( {- 1} \right)}}{{3 - 0}}\\&= \frac{1}{3}\\\end{aligned}$

Substitute $\dfrac{1}{3}$ for m, $3$ for $x$ and $0$ for $y$ in equation $y = mx + c$ to obtain the value of $c$.

$\begin{aligned}0& \frac{1}{3} \cdot 3 + c\\- 1&=c\\\end{aligned}$

To check whether the equation includes origin substitute 0 for x and 0 for y in equation $y \geqslant \dfrac{1}{3}x - 1.$

$\begin{aligned}0&\geqslant \frac{1}{3} \times 0 - 1 \\0&\geqslant - 1\\\end{gathered}$

The statement is true. The equation contains origin.

The equation of the line is $\boxed{y \geqslant \frac{1}{3}x - 1}.$

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Answer details:

Grade: High School

Subject: Mathematics

Chapter: Linear equation

Keywords: numbers,slope, slope intercept, inequality, equation, linear inequality, shaded region, y-intercept, graph, representation, origin.

Answer 2

Linear inequality represented by the graph is

$\displaystyle (A)y\geq \dfrac{1}{3}x-1$

Further explanation

Straight-line equations are mathematical equations that are described in the plane of cartesian coordinates

General formula

 $\large{\boxed{\bold{y-y_1=m(x-x_1)}}$

or

y = mx + c

Where

m = straight-line gradient which is the slope of the line

x1, y1 = the Cartesian coordinate that is crossed by the line

c = constant

The formula for a gradient (m) between 2 points  

$\large{\boxed{\bold{m=\dfrac{y_2-y_1}{x_2-x_1}}}$

If the intersection of the x-axis (b, 0) and the y-axis (0, a) then the equation of the line:

ax + by = ab

It says inequality if there are symbol forms like <, >, ≤ or ≥

Whereas linear inequality can have forms:  

ax + by> c, ax + by ≥ c , ax + by <c , ax + by ≤ c

In graphical form, line inequality can be

• dashed line because y does not include equals to

• a solid line because y includes equal to

For line inequality (positive coefficient y)

ax + by ≥ c then the solution is shaded upwards

ax + by ≤ c then the solution is shaded down

Or we input the values ​​x, y from the point in the shaded area and put in the inequality line

From the picture we can determine the equation of the line

Line through 2 points (3,0) and (0,-1)

the gradient:

$\displaystyle m=\frac{-1-0}{0-3}\\\\\displaystyle m=\frac{1}{3}$

the equation of the line:  point (3,0)

$\displaystyle y-0=\frac{1}{3}(x-3)\\\\\displaystyle y=\frac{1}{3}x-1$

We check the point in the area of ​​shading, for example (0, 0)

we input in the equation :

$\displaystyle 0=\frac{1}{3}.0-1\\\\0=-1$

Because 0 > -1 and the graph is solid line so the inequality line will be

$\displaystyle (A)y\geq \dfrac{1}{3}x-1$

 

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Keywords: linear inequality,graph