# 4.7 Graphs of linear inequalities  (Page 3/10)

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The boundary line shown is $2x+3y=6$ . Write the inequality shown by the graph.

The line $2x+3y=6$ is the boundary line. On one side of the line are the points with $2x+3y>6$ and on the other side of the line are the points with $2x+3y<6$ .

Let’s test the point $\left(0,0\right)$ and see which inequality describes its side of the boundary line.

At $\left(0,0\right)$ , which inequality is true:

$\begin{array}{ccccccccccc}\hfill 2x+3y& >\hfill & 6\hfill & & & \hfill \text{or}\hfill & & & \hfill 2x+3y& <\hfill & 6?\hfill \\ \hfill 2x+3y& >\hfill & 6\hfill & & & & & & \hfill 2x+3y& <\hfill & 6\hfill \\ \hfill 2\left(0\right)+3\left(0\right)& \stackrel{?}{>}\hfill & 6\hfill & & & & & & \hfill 2\left(0\right)+3\left(0\right)& \stackrel{?}{<}\hfill & 6\hfill \\ \hfill 0& >\hfill & 6\phantom{\rule{0.2em}{0ex}}\text{False}\hfill & & & & & & \hfill 0& <\hfill & 6\phantom{\rule{0.2em}{0ex}}\text{True}\hfill \end{array}$

So the side with $\left(0,0\right)$ is the side where $2x+3y<6$ .

(You may want to pick a point on the other side of the boundary line and check that $2x+3y>6$ .)

Since the boundary line is graphed as a dashed line, the inequality does not include an equal sign.

The graph shows the solution to the inequality $2x+3y<6$ .

Write the inequality shown by the shaded region in the graph with the boundary line $x-4y=8$ .

$x-4y\le 8$

Write the inequality shown by the shaded region in the graph with the boundary line $3x-y=6$ .

$3x-y\le 6$

## Graph linear inequalities

Now, we’re ready to put all this together to graph linear inequalities.

## How to graph linear inequalities

Graph the linear inequality     $y\ge \frac{3}{4}x-2$ .

## Solution

Graph the linear inequality $y\ge \frac{5}{2}x-4$ .

Graph the linear inequality $y<\frac{2}{3}x-5$ .

The steps we take to graph a linear inequality are summarized here.

## Graph a linear inequality.

1. Identify and graph the boundary line.
• If the inequality is $\le \text{or}\ge$ , the boundary line is solid.
• If the inequality is<or>, the boundary line is dashed.
2. Test a point that is not on the boundary line. Is it a solution of the inequality?
3. Shade in one side of the boundary line.
• If the test point is a solution, shade in the side that includes the point.
• If the test point is not a solution, shade in the opposite side.

Graph the linear inequality $x-2y<5$ .

## Solution

First we graph the boundary line $x-2y=5$ . The inequality is $<$ so we draw a dashed line.

Then we test a point. We’ll use $\left(0,0\right)$ again because it is easy to evaluate and it is not on the boundary line.

Is $\left(0,0\right)$ a solution of $x-2y<5$ ?

The point $\left(0,0\right)$ is a solution of $x-2y<5$ , so we shade in that side of the boundary line.

Graph the linear inequality $2x-3y\le 6$ .

Graph the linear inequality $2x-y>3$ .

What if the boundary line goes through the origin? Then we won’t be able to use $\left(0,0\right)$ as a test point. No problem—we’ll just choose some other point that is not on the boundary line.

Graph the linear inequality $y\le -4x$ .

## Solution

First we graph the boundary line $y=-4x$ . It is in slope–intercept form, with $m=-4\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b=0$ . The inequality is $\le$ so we draw a solid line.

Now, we need a test point. We can see that the point $\left(1,0\right)$ is not on the boundary line.

Is $\left(1,0\right)$ a solution of $y\le -4x$ ?

The point $\left(1,0\right)$ is not a solution to $y\le -4x$ , so we shade in the opposite side of the boundary line. See [link] .

Graph the linear inequality $y>-3x$ .

Graph the linear inequality $y\ge -2x$ .

Some linear inequalities have only one variable. They may have an x but no y , or a y but no x . In these cases, the boundary line will be either a vertical or a horizontal line. Do you remember?

$\begin{array}{cccc}x=a\hfill & & & \text{vertical line}\hfill \\ y=b\hfill & & & \text{horizontal line}\hfill \end{array}$

Graph the linear inequality $y>3$ .

## Solution

First we graph the boundary line $y=3$ . It is a horizontal line. The inequality is>so we draw a dashed line.

We test the point $\left(0,0\right)$ .

$\begin{array}{}\\ y>3\hfill \\ \\ 0\overline{)>}3\hfill \end{array}$

$\left(0,0\right)$ is not a solution to $y>3$ .

So we shade the side that does not include (0, 0).

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Mckenzie
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Mr Hernaez runs his car at a regular speed of 50 kph and Mr Ranola at 36 kph. They started at the same place at 5:30 am and took opposite directions. At what time were they 129 km apart?
90 minutes
Melody wants to sell bags of mixed candy at her lemonade stand. She will mix chocolate pieces that cost $4.89 per bag with peanut butter pieces that cost$3.79 per bag to get a total of twenty-five bags of mixed candy. Melody wants the bags of mixed candy to cost her $4.23 a bag to make. How many bags of chocolate pieces and how many bags of peanut butter pieces should she use? Jake Reply enrique borrowed$23,500 to buy a car he pays his uncle 2% interest on the $4,500 he borrowed from him and he pays the bank 11.5% interest on the rest. what average interest rate does he pay on the total$23,500
13.5
Pervaiz
Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tiles. She will use basic tiles that cost $8 per square foot and decorator tiles that cost$20 per square foot. How many square feet of each tile should she use so that the overall cost of the backsplash will be $10 per square foot? Bridget Reply The equation P=28+2.54w models the relation between the amount of Randy’s monthly water bill payment, P, in dollars, and the number of units of water, w, used. Find the payment for a month when Randy used 15 units of water. Bridget help me understand graphs Marlene Reply what kind of graphs? bruce function f(x) to find each value Marlene I am in algebra 1. Can anyone give me any ideas to help me learn this stuff. Teacher and tutor not helping much. Marlene Given f(x)=2x+2, find f(2) so you replace the x with the 2, f(2)=2(2)+2, which is f(2)=6 Melissa if they say find f(5) then the answer would be f(5)=12 Melissa I need you to help me Melissa. Wish I can show you my homework Marlene How is f(1) =0 I am really confused Marlene what's the formula given? f(x)=? Melissa It shows a graph that I wish I could send photo of to you on here Marlene Which problem specifically? Melissa which problem? Melissa I don't know any to be honest. But whatever you can help me with for I can practice will help Marlene I got it. sorry, was out and about. I'll look at it now. Melissa Thank you. I appreciate it because my teacher assumes I know this. My teacher before him never went over this and several other things. Marlene I just responded. Melissa Thank you Marlene -65r to the 4th power-50r cubed-15r squared+8r+23 ÷ 5r WENDY Reply State the question clearly please Rich write in this form a/b answer should be in the simplest form 5% August Reply convert to decimal 9/11 August 0.81818 Rich 5/100 = .05 but Rich is right that 9/11 = .81818 Melissa Equation in the form of a pending point y+2=1/6(×-4) Jose Reply write in simplest form 3 4/2 August definition of quadratic formula Ahmed Reply From Google: The quadratic formula, , is used in algebra to solve quadratic equations (polynomial equations of the second degree). The general form of a quadratic equation is , where x represents a variable, and a, b, and c are constants, with . A quadratic equation has two solutions, called roots. Melissa what is the answer of w-2.6=7.55 What Reply 10.15 Michael w = 10.15 You add 2.6 to both sides and then solve for w (-2.6 zeros out on the left and leaves you with w= 7.55 + 2.6) Korin Nataly is considering two job offers. The first job would pay her$83,000 per year. The second would pay her $66,500 plus 15% of her total sales. What would her total sales need to be for her salary on the second offer be higher than the first? Mckenzie Reply x >$110,000
bruce
greater than $110,000 Michael Estelle is making 30 pounds of fruit salad from strawberries and blueberries. Strawberries cost$1.80 per pound, and blueberries cost $4.50 per pound. If Estelle wants the fruit salad to cost her$2.52 per pound, how many pounds of each berry should she use?
$1.38 worth of strawberries +$1.14 worth of blueberries which= $2.52 Leitha how Zaione is it right😊 Leitha lol maybe Robinson 8 pound of blueberries and 22 pounds of strawberries Melissa 8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound Melissa 8 pounds x 4.5 equal 36 22 pounds x 1.80 equal 39.60 36 + 39.60 equal 75.60 75.60 / 30 equal average 2.52 per pound Melissa hmmmm...... ? Robinson 8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound Melissa The question asks how many pounds of each in order for her to have an average cost of$2.52. She needs 30 lb in all so 30 pounds times $2.52 equals$75.60. that's how much money she is spending on the fruit. That means she would need 8 pounds of blueberries and 22 lbs of strawberries to equal 75.60
Melissa
good
Robinson
👍
Leitha
thanks Melissa.
Leitha
nawal let's do another😊
Leitha
we can't use emojis...I see now
Leitha
Sorry for the multi post. My phone glitches.
Melissa