Solve the inequality both algebraically and graphically. Draw a number line graph of the solution and give interval notation.

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asked May 18, 2023 45.0k views

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James Nguyen · Answer 1 · 2023-05-22T08:39:23+0000

Finding the solution algebraically

To answer this inequality, we can follow the next steps:

1. Multiply by 7 both sides of the inequality:

$7\cdot((x-7))/(2)<7\cdot(41)/(7)\Rightarrow7\cdot((x-7))/(2)<41$

2. Multiply by 2 both sides of the inequality:

$7\cdot2\cdot^(\cdot)((x-7))/(2)<2\cdot41\Rightarrow7\cdot(x-7)<82$

3. Apply the distributive property at the left side of the inequality:

$7\cdot x-7\cdot7<82\Rightarrow7x-49<82$

4. Add 49 to both sides of the inequality:

$7x-49+49<82+49\Rightarrow7x<131$

5. Finally, divide both sides of the inequality by 7:

$(7x)/(7)<(131)/(7)\Rightarrow x<(131)/(7)$

We can graph this inequality in the number line as follows:

Notice the parenthesis indicating that the solution is the number below 131/7 (but not equal to 131/7). In interval notation the solution is:

$(-\infty,(131)/(7))$
$(-\infty,18(5)/(7))$

Or, approximately:

$(-\infty,18.7142857143)$

The number 131/7 in decimal is equivalent to 18.7142857143, so the graph of the solution is given by graph A (we can see that there are seven divisions between 18 and 19; since we have that the shaded division is in the 5th division, then, we have 5/7 = 0.714285714286, that is, the decimal part of the above number).

We can express the number 131/7 as a mixed number as follows:

(131)/(7)=(126)/(7)+(5)/(7)=18+(5)/(7)=18(5)/(7)

Again, notice also the symbol for the left part of the interval notation is a parenthesis since the interval is open at the point 131/7 = 18 + 5/7.

Finding the solution graphically

To find the solution graphically, we can represent the inequality as two lines as follows:

Then, if we graph the first line, we can find the x- and the y-intercepts to find two points to graph the line. We have that the x- and the y-intercepts are:

The x-intercept is (that is, when y = 0):

$0=(x-7)/(2)\Rightarrow x-7=0\Rightarrow x=7$

Then, the x-intercept is (7, 0), and the y-intercept (the point on the graph when x = 0) is:

$y=(x-7)/(2)\Rightarrow y=(0-7)/(2)\Rightarrow y=-(7)/(2)$

Then, the y-intercept is (0, -7/2).

The other line is given by:

With this information, we can graph both lines:

And we can see that the point where the two lines coincide is:

Then, the values for x of the line (x-7)/2 [that is, the values of y = (x-7)/2] that are less than y = 41/7, represented as:

Are those values of x less than 131/7, or the solution is also (we express the solution as a fraction or a mixed number as follows) (the same solution):

$(-\infty,(131)/(7))or(-\infty,18(5)/(7))$

In summary, we have that the solution to the inequality is:

As an inequality: