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SAT Math : Equations / Inequalities

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Example Questions

Example Question #1 : Inequalities

$\dpi{100} \small -2y+7>-7+y$

Given the inequality above, which of the following MUST be true?

Possible Answers:

$\dpi{100} \small y>\frac{14}{3}$

$\dpi{100} \small y>-5$ $y < \textup{5}$

$\dpi{100} \small y<\frac{-14}{3}$

$\dpi{100} \small y<5$ $y< \textup{3}$

$\dpi{100} \small y>\frac{-14}{3}$ $y>\textup{2}$

Correct answer:

$\dpi{100} \small y>-5$ $y < \textup{5}$

Explanation:

$\dpi{100} \small -2y+7>-7+y$ Subtract from both sides:

-2y+7-y>-7+y-y

-3y+7>-7

Subtract 7 from both sides:

-3y+7-7>-7-7

-3y>-14

Divide both sides by $\dpi{100} \small -3$ :

$\frac{-3y}{-3}>\frac{-14}{-3}$

Remember to switch the inequality when dividing by a negative number:

$y<\frac{14}{3}$

Since $\dpi{100} \small y<\frac{14}{3}$ is not an answer, we must find an answer that, at the very least, does not contradict the fact that is less than (approximately) 4.67. Since any number that is less than 4.67 is also less than any number that is bigger than 4.67, we can be sure that is less than 5.

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Example Question #2 : Inequalities

A factory packs cereal boxes. Before sealing each box, a machine weighs it to ensure that it is no lighter than 356 grams and no heavier than 364 grams. If the box holds grams of cereal, which inequality represents all allowable values of ?

Possible Answers:

$\left | w+360 \right |<4$

$\left | w+360 \right |\geq4$

$\left | w-360 \right |>4$

$\left | w-360 \right |=4$

$\left | w-360 \right |\leq4$

Correct answer:

$\left | w-360 \right |\leq4$

Explanation:

The median weight of a box of cereal is 360 grams. This should be an allowable value of w. Substituting 360 for w into each answer choice, the only true results are:

$\\ |w -360| \leq 4 \\ |360 -360| \leq 4 \\0 \leq 4$

and:

$\\|w + 360| \geq 4 \\ |360 + 360| \geq 4\\ 720 \geq4$

Notice that any positive value for w satisfies the second inequality above. Since w must be between 356 and 364, the first inequality above is the only reasonable choice.

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Example Question #11 : How To Find The Solution To An Inequality With Subtraction

Solve for .

$4x-8\geq 28$

Possible Answers:

x<9

x>9

x=9

$x\leq 9$

$x\geq 9$

Correct answer:

$x\geq 9$

Explanation:

We want to isolate the variable on one side and numbers on another side. Treat like a normal equation.

$4x-8\geq 28$ Add on both sides.

$4x\geq 36$ Divide on both sides.

$x\geq 9$

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Example Question #13 : Inequalities

Solve for .

-9x-38>25

Possible Answers:

x>-7

x<-7

$x\geq -7$

$x\leq -7$

x=-7

Correct answer:

x<-7

Explanation:

We want to isolate the variable on one side and numbers on another side. Treat like a normal equation.

-9x-38>25 Add on both sides.

-9x>63 Divide on both sides. Remember to flip the sign.

x<-7

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Example Question #13 : How To Find The Solution To An Inequality With Subtraction

Solve for .

7x-27<9x-77

Possible Answers:

x>-25

x>25

x<25

x>50

x<50

Correct answer:

x>25

Explanation:

We want to isolate the variable on one side and numbers on another side. Treat like a normal equation.

7x-27<9x-77 Add and subtract on both sides.

50<2x Divide on both sides.

25<x

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Example Question #14 : How To Find The Solution To An Inequality With Subtraction

Solve for .

$\left | 6x-45\right |\leq 75$

Possible Answers:

$5\leq x\leq 20$

$-5\leq x\leq 20$

$x\leq 20$

$-5\leq x$

$x\geq 20, x\leq -5$

Correct answer:

$-5\leq x\leq 20$

Explanation:

We want to isolate the variable on one side and numbers on another side. Treat like a normal equation.

$\left | 6x-45\right |\leq 75$ We need to set-up two equations since its absolute value.

$6x-45\leq 75$ Add on both sides.

$6x\leq 120$ Divide on both sides.

$x\leq 20$

$-(6x-45)\leq 75$ Divide on both sides which flips the sign.

$6x-45\geq -75$ Add on both sides.

$6x\geq -30$ Divide on both sides.

$x\geq -5$

Since we have the 's being either greater than or less than the values, we can combine them to get $-5\leq x\leq 20$ .

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Example Question #16 : Inequalities

Solve for .

$\left | 5x-75\right |>45-x$

Possible Answers:

7.5<x<20

x>20

x>7.5

-20<x<-7.5

x>7.5, x>20

Correct answer:

x>20

Explanation:

We want to isolate the variable on one side and numbers on another side. Treat like a normal equation.

$\left | 5x-75\right |>45-x$ We need to set-up two equations since it's absolute value.

5x-75>45-x Add x, 75 on both sides.

6x>120 Divide on both sides.

x>20

-(5x-75)>45-x Distribute the negative sign to each term in the parenthesis.

-5x+75>45-x Add and subtract on both sides.

30>4x Divide on both sides.

7.5>x We must check each answer. Let's try .

$\left | 10(5)-75\right |>45-10$ 25>35 This is not true therefore 7.5>x is not correct. Let's try .

$\left | 5(30)-75\right |>45-30$ 75>15 This true so therefore x>20 is correct.

Final answer is x>20 .

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Example Question #15 : How To Find The Solution To An Inequality With Subtraction

Solve for .

$\left | x-19\right |<2x-26$

Possible Answers:

7<x

15<x

x< 15

7<x, 15<x

7<x<15

Correct answer:

15<x

Explanation:

We want to isolate the variable on one side and numbers on another side. Treat like a normal equation.

$\left | x-19\right |<2x-26$ We need to set-up two equations since it's absolute value.

x-19<2x-26 Subtract and add on both sides.

7<x

-(x-19)<2x-26 Distribute the negative sign to each term in the parenthesis.

-x+19<2x-26 Add , on both sides.

45<3x Divide on both sides.

15<x We must check each answer. Let's try .

$\left | 10-19\right |<2(10)-26$ 9<-6 This is not true therefore 7<x is not correct. Let's try .

$\left | 20-19\right |<2(20)-26$ 1<20 This true so therefore 15<x is correct.

Final answer is 15<x .

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Example Question #5 : Inequalities

What values of x make the following statement true?

|x – 3| < 9

Possible Answers:

–6 < x < 12

x < 12

–3 < x < 9

–12 < x < 6

6 < x < 12

Correct answer:

–6 < x < 12

Explanation:

Solve the inequality by adding 3 to both sides to get x < 12. Since it is absolute value, x – 3 > –9 must also be solved by adding 3 to both sides so: x > –6 so combined.

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Example Question #1 : How To Find The Solution To An Inequality With Addition

If –1 < w < 1, all of the following must also be greater than –1 and less than 1 EXCEPT for which choice?

Possible Answers:

|w|^0.5

w/2

3w/2

w²

|w|

Correct answer:

3w/2

Explanation:

3w/2 will become greater than 1 as soon as w is greater than two thirds. It will likewise become less than –1 as soon as w is less than negative two thirds. All the other options always return values between –1 and 1.

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SAT Math : Equations / Inequalities

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #1 : Inequalities

Example Question #2 : Inequalities

Example Question #11 : How To Find The Solution To An Inequality With Subtraction

Example Question #13 : Inequalities

Example Question #13 : How To Find The Solution To An Inequality With Subtraction

Example Question #14 : How To Find The Solution To An Inequality With Subtraction

Example Question #16 : Inequalities

Example Question #15 : How To Find The Solution To An Inequality With Subtraction

Example Question #5 : Inequalities

Example Question #1 : How To Find The Solution To An Inequality With Addition

All SAT Math Resources

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