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linear inequality graph 2 - variables which inequality is shown on the …

Question

linear inequality graph 2 - variables
which inequality is shown on the graph?
a ( y < 4x + 5 )
b ( y < - 4x )
c ( y < - 4x + 5 )
d ( y < - \frac{1}{4}x + 5 )
which ordered pair is not a solution to the inequality?
a ( (1, 1) )
b ( (0, 2) )
c ( (1, - 2) )
d ( (- 2, - 2) )

Explanation:

Response
First Sub - Question: Which inequality is shown on the graph?
Step - by - Step Explanation:
Step 1: Find the slope of the line

The general form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept.
We can see that the line passes through the points \((0,5)\) (the y - intercept, so \(b = 5\)) and \((1,1)\) (we can also use other points). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's take \((0,5)\) as \((x_1,y_1)\) and \((1,1)\) as \((x_2,y_2)\). Then \(m=\frac{1 - 5}{1-0}=\frac{- 4}{1}=-4\).

Step 2: Determine the inequality sign and the equation

The line is dashed, which means the inequality is either \(y\lt mx + b\) or \(y\gt mx + b\). The shaded region is to the left of the line. Let's test a point in the shaded region, say \((0,0)\).
For option A: \(y\lt4x + 5\). Substitute \(x = 0,y = 0\): \(0\lt4(0)+5\), \(0\lt5\) is true, but the slope of the line in the graph is - 4, not 4. So A is wrong.
For option B: \(y\lt - 4x\). Substitute \(x = 0,y = 0\): \(0\lt-4(0)\), \(0\lt0\) is false. And the y - intercept of the line in the graph is 5, not 0. So B is wrong.
For option C: \(y\lt - 4x+5\). Substitute \(x = 0,y = 0\): \(0\lt-4(0)+5\), \(0\lt5\) is true. The slope \(m=-4\) and y - intercept \(b = 5\) match our earlier calculations.
For option D: \(y\lt-\frac{1}{4}x + 5\). The slope of the line in the graph is - 4, not \(-\frac{1}{4}\). So D is wrong.
The equation of the line is \(y=-4x + 5\) and the shaded region satisfies \(y\lt - 4x+5\).

Second Sub - Question: Which ordered pair is NOT a solution to the inequality \(y\lt - 4x + 5\)?
Step - by - Step Explanation:

We will substitute each ordered pair into the inequality \(y\lt - 4x+5\)

Step 1: Test option A: \((1,1)\)

Substitute \(x = 1,y = 1\) into \(y\lt - 4x + 5\).
\(1\lt-4(1)+5\)
\(1\lt-4 + 5\)
\(1\lt1\) is false. Wait, let's recalculate: \(-4(1)+5=-4 + 5 = 1\). So \(1\lt1\) is false? Wait, no, the inequality is \(y\lt - 4x+5\). Wait, maybe I made a mistake. Let's check again.
Wait, the inequality from the first question is \(y\lt - 4x + 5\)
For option A: \((1,1)\)
Left - hand side (LHS): \(y = 1\)
Right - hand side (RHS): \(-4(1)+5=-4 + 5 = 1\)
Since \(1\lt1\) is false, but wait, maybe I made a mistake in the first question? Wait, no, let's check the other options.
For option B: \((0,2)\)
Substitute \(x = 0,y = 2\) into \(y\lt - 4x+5\)
\(2\lt-4(0)+5\)
\(2\lt5\) is true.
For option C: \((1,-2)\)
Substitute \(x = 1,y=-2\) into \(y\lt - 4x + 5\)
\(-2\lt-4(1)+5\)
\(-2\lt1\) is true.
For option D: \((-2,-2)\)
Substitute \(x=-2,y = - 2\) into \(y\lt - 4x+5\)
\(-2\lt-4(-2)+5\)
\(-2\lt8 + 5\)
\(-2\lt13\) is true.

Wait, for the first sub - question, when we calculated the slope, we can also see from the graph that the line goes from \((0,5)\) to \((1,1)\), so the slope \(m=\frac{1 - 5}{1-0}=-4\), and the y - intercept \(b = 5\), and the shaded region is below the line (since the line is dashed and the shading is towards the left - down), so the inequality is \(y\lt - 4x + 5\), so the answer for the first sub - question is C. For the second sub - question, when we substitute \((1,1)\) into \(y\lt - 4x + 5\), we get \(1\lt-4\times1 + 5=1\), which is false. But wait, maybe there is a miscalculation. Wait, \(-4\times1+5 = 1\), so \(1\lt1\) is false. But let's check the other points again.
Wait, for the point \((1,1)\):
The inequality is \(y\lt - 4x+5\). So \(1\lt-4(1)+5\)
\(-4(1)+5 = 1\), so \(1\lt1\) is not true. For the point \((0,2)\): \(2\lt-4(0)+5=5\), which is true. For \((1,-2)\): \(-2\lt-4(1)+5 = 1\),…

Answer:

s:

  • For the first sub - question: C. \(y\lt - 4x + 5\)
  • For the second sub - question: A. \((1,1)\)