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SAT Math : How to graph an inequality with a number line

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Example Questions

Example Question #1 : Number Lines And Absolute Value

On a real number line, x₁ = -4 and x₂ = 14. What is the distance between these two points?

Possible Answers:

-18

Correct answer:

Explanation:

The distance between two points is always positive. We calculate lx₂ - x₁l, which will give us the distance between the points.

|14- (-4)| = |14+4| = |18| = 18

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Example Question #2 : Number Line

$26 \le 2x <64$ $\displaystyle 26 \le 2x < 64$

Which of the following is a graph for the values of $\displaystyle x$ defined by the inequality stated above?

Possible Answers:

Ineq25

Ineq22

Ineq24

Ineq21

Ineq23

Correct answer:

Ineq21

Explanation:

To begin, you must simplify so that you "isolate" $\displaystyle x$, (i.e. at least eliminate any coefficients from it). To do this, divide all of the members of the inequality by $\displaystyle 2$:

$13 \le x <32$ $\displaystyle 13 \le x < 32$

Now, this inequality represents all of the numbers between 13 and 32. However, it does include $\displaystyle 13$ (hence, getting a closed circle for that value) and does not include $\displaystyle 32$ (hence, getting an open circle for that value). Therefore, it looks like:

Ineq21

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Example Question #1 : Graphing An Inequality With A Number Line

Ineq3

Which of the following inequalities is represented by the number line shown above?

Possible Answers:

$4x \geq -8$ $\displaystyle 4x \geq -8$

$| 5x - 5 | \geq 25$ $\displaystyle | 5x - 5 | \geq 25$

$|4x - 6| \leq 14$ $\displaystyle |4x - 6| \leq 14$

$|3x - 12| \leq 6$ $\displaystyle |3x - 12| \leq 6$

$5x \leq 25$ $\displaystyle 5x \leq 25$

Correct answer:

$|4x - 6| \leq 14$ $\displaystyle |4x - 6| \leq 14$

Explanation:

Since the inequality represents one range of values between two end points (both of which are included, given the sign being "less than or equal"), you know that whatever you answer, it must be convertible to the form:

$-2 \leq x \leq 5$ $\displaystyle -2 \leq x \leq 5$

Now, you know that it is impossible to get this out of the choices that have no absolute values involved in them. Therefore, the only options that make sense are the two having absolute values; however, here you should choose only the ones that have a $\leq$ $\displaystyle \leq$, for only that will yield a range like this. Thus, we can try both of our options.

The wrong answer is simplified in this manner:

$|3x - 12| \leq 6$ $\displaystyle |3x - 12| \leq 6$

$-6\leq 3x - 12 \leq 6$ $\displaystyle -6\leq 3x - 12 \leq 6$

$6\leq 3x \leq 18$ $\displaystyle 6\leq 3x \leq 18$

And you can stop right here, for you know you will never have $\displaystyle -2$ for the left terminus.

The other option is simplified in this manner:

$|4x - 6| \leq 14$ $\displaystyle |4x - 6| \leq 14$

$-14 \leq 4x - 6 \leq 14$ $\displaystyle -14 \leq 4x - 6 \leq 14$

$-8 \leq 4x \leq 20$ $\displaystyle -8 \leq 4x \leq 20$

$-2 \leq x \leq 5$ $\displaystyle -2 \leq x \leq 5$

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Example Question #1 : Number Line

$\displaystyle 2x + 6 > 16$

Which of the following is a graph for the values of $\displaystyle x$ defined by the inequality stated above?

Possible Answers:

Ineq13

Ineq12

Ineq11

Ineq14

Ineq15

Correct answer:

Ineq11

Explanation:

Begin by solving for $\displaystyle x$:

$\displaystyle 2x > 16-6$

2x > 10 $\displaystyle 2x > 10$

x> 5 $\displaystyle x> 5$

Now, this is represented by drawing an open circle at 6 and graphing upward to infinity:

Ineq11

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SAT Math : How to graph an inequality with a number line

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #1 : Number Lines And Absolute Value

Example Question #2 : Number Line

Example Question #1 : Graphing An Inequality With A Number Line

Example Question #1 : Number Line

All SAT Math Resources

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