Which of the following ordered pairs is a solution to the system of inequalities above?

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SAT Linear Inequalities Question 95 | Free SAT Prep | Aniko

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Question 95·Medium·Linear Inequalities in One or Two Variables

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\begin{aligned} y &< 2x - 1 \\ x &\ge -1 \end{aligned}

Which of the following ordered pairs $(x, y)$ is a solution to the system of inequalities above?

For systems of inequalities with a small set of answer choices, the fastest method is usually to plug in each option rather than graphing. First, use any simple inequality (like one involving just $x$ or just $y$ ) to quickly eliminate choices. Then, for the remaining options, substitute their coordinates into the more complicated inequality, being careful with strict symbols like $<$ versus $\le$ . This plug-in-and-check approach is quick, avoids algebra mistakes, and works very well under time pressure on the SAT.

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Think about what a solution to a system means

For a point $(x,y)$ to be a solution, what has to be true about both inequalities when you plug in $x$ and $y$ ?

Start with the easier inequality

Look at the inequality $x\ge -1$ and compare it to the $x$ -values in the answer choices. Can you immediately rule out any choices based on this?

Substitute into the other inequality

For the points that are not ruled out, substitute each $x$ and $y$ into $y<2x-1$ . Carefully compare the left and right sides: is the left side actually less than the right side?

Pay attention to strict vs. non-strict inequalities

Remember that $<$ means strictly less than. If both sides are equal, does that satisfy $<$ ?

Desmos Guide

Graph the inequalities

In Desmos, type y < 2x - 1 and x >= -1. Desmos will shade the region that satisfies both inequalities (the overlap of the shaded areas).

Plot the answer choice points

Enter each answer choice as a point: (-1,-3), (0,-3), (2,4), and (-2,-5). Desmos will show each point on the graph.

Check which point lies in the solution region

Look at the graph and see which of the plotted points lies inside the overlapping shaded region that represents the solutions to both inequalities. The point that lies in this region corresponds to the correct answer choice.

Step-by-step Explanation

Understand what it means to be a solution

A point $(x,y)$ is a solution to a system of inequalities if it makes every inequality in the system true at the same time.

Here, a point must satisfy both:

$y<2x-1$
$x\ge -1$

Use the inequality $x \ge -1$ to filter choices

Look at the $x$ -values in each answer choice and see if they are at least $-1$ .

$(-1,-3)$ has $x=-1$ , which satisfies $x\ge -1$ .
$(0,-3)$ has $x=0$ , which also satisfies $x\ge -1$ .
$(2,4)$ has $x=2$ , which satisfies $x\ge -1$ .
$(-2,-5)$ has $x=-2$ , which does not satisfy $x\ge -1$ .

So $(-2,-5)$ cannot be a solution. You only need to test the other three points in the remaining inequality $y<2x-1$ .

Test each remaining point in $y<2x-1$

Substitute each remaining point into $y<2x-1$ and check if the inequality is true.

For $(-1,-3)$ :
- Left side: $y=-3$
- Right side: $2x-1=2(-1)-1=-2-1=-3$
- Compare: is $-3<-3$ ? No, this is false because $-3$ is equal to $-3$ , not less.
For $(0,-3)$ :
- Left side: $y=-3$
- Right side: $2x-1=2(0)-1=0-1=-1$
- Compare: is $-3<-1$ ? Yes, this is true.
For $(2,4)$ :
- Left side: $y=4$
- Right side: $2x-1=2(2)-1=4-1=3$
- Compare: is $4<3$ ? No, this is false.

So only one of these points makes $y<2x-1$ true.

Combine both inequalities to choose the answer

A correct solution must satisfy both $x\ge -1$ and $y<2x-1$ .

$(-1,-3)$ passes $x\ge -1$ but fails $y<2x-1$ .
$(2,4)$ passes $x\ge -1$ but fails $y<2x-1$ .
$(0,-3)$ passes $x\ge -1$ and makes $y<2x-1$ true.

Therefore, the ordered pair that is a solution to the system is $(0,-3)$ .

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation