Question

homework please help me understand the inequalities

Answer 1

Answer:

1st graph

$y\leq \frac{2}{3} x-2$

2nd graph

$y<3x$

3rd graph

$\left \{ {y<3x} \atop {y<-\frac{1}{2}x+4} \right.$

Step-by-step explanation:

1st graph

The inequality includes all the points on the line(As it is a solid line not a dotted line) and below that. Its below because the shaded part in under the line drawn.

You can find the expression for the line graph by using the general formula for a line graph:

y=mx+c

gradient/slope(m) of the graph is $\frac{2}{3}$ and intercept(c) is -2.

We can write the inequality as,

$y\leq \frac{2}{3} x-2$

2nd Graph

The inequality includes all the points on the line(As it is a solid line not a dotted line) and above that. Its above because the shaded part is above the line drawn.

Gradient/slope(m) of the graph is 0 as it is a horizontal line.

Intercept of the line is 2 as it intersects y axis at 2 when x=0.

Therefore the inequality can be written as:

$y\geq 2$

3rd graph

This is a system of linear inequalities.

Here we need to consider the area that is covered by both lines. (Where it has squares and a S written)

That area is under both the line that has a positive slope and a negative slope.

Let's consider the line with positive slope first.

Slope(m) = $\frac{3-0}{1-0}$=$3$

Intercept(c) = 0

Therefore, the inequality is,

$y<3x$

Let's consider the line with negative slope now,

Slope(m) = $\frac{4-0}{0-8}$=$-\frac{1}{2}$

Intercept(c) = 4

Therefore, the inequality is,

$y<-\frac{1}{2}x+4$

So the system inequalities for the 3rd graph is,

$\left \{ {y<3x} \atop {y<-\frac{1}{2}x+4} \right.$