Solve the system of inequalities: $y+2 x>3$ and $y \geq 3.5 x-5$
The first inequality, $y+2 x>3$, is $\square$ in slope-intercept form.
The first inequality, $y+2 x>3$, has a $\square$ boundary line.
The second inequality, $y \geq 3.5 x-5$, has a $\square$ boundary line.
Both inequalities have a solution set that is shaded $\square$ their boundary lines.
$\square$ is a point in the solution set of the system of inequalities.
Answer (1)
Rewrite the first inequality in slope-intercept form: -2x + 3"> y > − 2 x + 3 .
Identify the boundary line types: dashed for -2x + 3"> y > − 2 x + 3 and solid for y ≥ 3.5 x − 5 .
Determine the shading direction: above both boundary lines.
Find a point in the solution set: ( 2 , 3 ) , which satisfies both inequalities. The final answer is:
( 2 , 3 )
Explanation
Understanding the Problem We are given a system of two inequalities:
3"> y + 2 x > 3
y ≥ 3.5 x − 5
Our goal is to solve this system, which means finding all points ( x , y ) that satisfy both inequalities. We will also describe the solution set and find a point within it.
Rewriting the First Inequality First, let's rewrite the first inequality in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept. We have:
3"> y + 2 x > 3
Subtract 2 x from both sides:
-2x + 3"> y > − 2 x + 3
So, the first inequality in slope-intercept form is -2x + 3"> y > − 2 x + 3 .
Determining the Boundary Line for the First Inequality The first inequality, -2x + 3"> y > − 2 x + 3 , has a boundary line defined by the equation y = − 2 x + 3 . Since the inequality is strict (i.e., it uses the "> > sign and not ≥ ), the boundary line is dashed, indicating that points on the line are not included in the solution set.
Determining the Boundary Line for the Second Inequality The second inequality, y ≥ 3.5 x − 5 , has a boundary line defined by the equation y = 3.5 x − 5 . Since the inequality is non-strict (i.e., it uses the ≥ sign), the boundary line is solid, indicating that points on the line are included in the solution set.
Determining the Shading Direction For the first inequality, -2x + 3"> y > − 2 x + 3 , the solution set consists of all points ( x , y ) such that y is greater than − 2 x + 3 . This means the solution set is shaded above the dashed boundary line. For the second inequality, y ≥ 3.5 x − 5 , the solution set consists of all points ( x , y ) such that y is greater than or equal to 3.5 x − 5 . This means the solution set is shaded above the solid boundary line.
Finding a Point in the Solution Set To find a point in the solution set, we need a point that satisfies both inequalities. Let's find the intersection point of the two boundary lines:
y = − 2 x + 3 y = 3.5 x − 5
Setting the two expressions for y equal to each other:
− 2 x + 3 = 3.5 x − 5
Add 2 x to both sides:
3 = 5.5 x − 5
Add 5 to both sides:
8 = 5.5 x
Divide by 5.5 :
x = 5.5 8 = 55 80 = 11 16 ≈ 1.45
Now, substitute this value of x into either equation to find y . Using the first equation:
y = − 2 ( 11 16 ) + 3 = − 11 32 + 11 33 = 11 1 ≈ 0.09
So, the intersection point is approximately ( 1.45 , 0.09 ) . Since the boundary lines themselves are not part of the solution (due to the strict inequality in the first equation), we need to choose a point that is not on either line and is in the region where both inequalities are satisfied. Let's pick a point to the 'upper right' of the intersection point, say ( 2 , 3 ) .
Verifying the Point Let's check if the point ( 2 , 3 ) satisfies both inequalities:
-2x + 3"> y > − 2 x + 3
-2(2) + 3"> 3 > − 2 ( 2 ) + 3 -4 + 3"> 3 > − 4 + 3 -1"> 3 > − 1 (True)
y ≥ 3.5 x − 5
3 ≥ 3.5 ( 2 ) − 5 3 ≥ 7 − 5 3 ≥ 2 (True)
Since ( 2 , 3 ) satisfies both inequalities, it is a point in the solution set.
Final Answer The first inequality, 3"> y + 2 x > 3 , is -2x + 3"> y > − 2 x + 3 in slope-intercept form. The first inequality, 3"> y + 2 x > 3 , has a dashed boundary line. The second inequality, y ≥ 3.5 x − 5 , has a solid boundary line. Both inequalities have a solution set that is shaded above their boundary lines. ( 2 , 3 ) is a point in the solution set of the system of inequalities.
Examples
Systems of inequalities are used in various real-world applications, such as linear programming, where the goal is to optimize a linear objective function subject to a set of linear inequality constraints. For example, a company might want to maximize its profit given constraints on the amount of resources available, such as labor and materials. Another application is in economics, where systems of inequalities can be used to model consumer behavior and market equilibrium. For instance, a consumer's budget constraint and preferences can be represented as a system of inequalities, and the solution set represents the set of affordable and desirable consumption bundles. In engineering, systems of inequalities can be used to design structures that meet certain safety and performance requirements. For example, the stresses and strains in a bridge must satisfy certain inequalities to ensure that the bridge does not collapse under load.
