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PSAT Math : Systems of Inequalities

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Example Questions

Example Question #1 : Solve One Variable Linear Equations And Inequalities: Ccss.Math.Content.Hsa Rei.B.3

Solve the following inequality for $\uptext{x}$ , round your answer to the nearest tenth.

$\sqrt{x + 8} > x + 1$

Possible Answers:

-3.2 < x < 2.2

x = 2.2

$-3.2 \leq x$

2.2 < x

2.2 < x < -3.2

Correct answer:

-3.2 < x < 2.2

Explanation:

The first step is to square each side of the inequality.

$\left( \sqrt{x + 8} \right)^2> \left( x + 1 \right)^2$

Now simplify each side.

$x + 8 > x^{2} + 2 x + 1$

Now subtract the left side of the inequality to make it zero, so that we can use the quadratic formula.

$x + 8 -\left( x + 8 \right)> x^{2} + 2 x + 1 -\left( x + 8 \right)$

$0 > x^{2} + x - 7$

Now we can use the quadratic formula.

Recall the quadratic formula.

$\frac{ -b \pm \sqrt{ b ^2 -4 \cdot a \cdot c }}{ 2 \cdot a }$

Where $\uptext{a}$ , $\uptext{b}$ , and $\uptext{c}$ , correspond to coefficients in the quadratic equation.

$a x^{2} + b x + c = 0$

In this case a = 1 , b = 1 , and c = -7 .

Now plug these values into the quadratic equation, and we get.

x = 2.2

x = -3.2

Now since we are dealing with an inequality, we put the least value on the left side, and the greatest value on the right. It will look like the following.

-3.2 < x < 2.2

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Example Question #491 : High School: Algebra

Solve the following inequality for $\uptext{x}$ , round your answer to the nearest tenth.

$\sqrt{x + 15} > x + 1$

Possible Answers:

3.3 < x < -4.3

x = 3.3

3.3 < x

-4.3 < x < 3.3

$-4.3 \leq x$

Correct answer:

-4.3 < x < 3.3

Explanation:

The first step is to square each side of the inequality.

$\left( \sqrt{x + 15} \right)^2> \left( x + 1 \right)^2$

Now simplify each side.

$x + 15 > x^{2} + 2 x + 1$

Now subtract the left side of the inequality to make it zero, so that we can use the quadratic formula.

$x + 15 -\left( x + 15 \right)> x^{2} + 2 x + 1 -\left( x + 15 \right)$

$0 > x^{2} + x - 14$

Now we can use the quadratic formula.

Recall the quadratic formula.

$\frac{ -b \pm \sqrt{ b ^2 -4 \cdot a \cdot c }}{ 2 \cdot a }$

Where $\uptext{a}$ , $\uptext{b}$ , and $\uptext{c}$ , correspond to coefficients in the quadratic equation.

$a x^{2} + b x + c = 0$

In this case a = 1 , b = 1 , and c = -14 .

Now plug these values into the quadratic equation, and we get.

x = 3.3

x = -4.3

Now since we are dealing with an inequality, we put the least value on the left side, and the greatest value on the right. It will look like the following.

-4.3 < x < 3.3

Report an Error

Example Question #191 : Equations / Inequalities

Solve the following inequality for $\uptext{x}$ , round your answer to the nearest tenth.

$\sqrt{x + 16} > x + 1$

Possible Answers:

3.4 < x < -4.4

$-4.4 \leq x$

-4.4 < x < 3.4

3.4 < x

x = 3.4

Correct answer:

-4.4 < x < 3.4

Explanation:

The first step is to square each side of the inequality.

$\left( \sqrt{x + 16} \right)^2> \left( x + 1 \right)^2$

Now simplify each side.

$x + 16 > x^{2} + 2 x + 1$

Now subtract the left side of the inequality to make it zero, so that we can use the quadratic formula.

$x + 16 -\left( x + 16 \right)> x^{2} + 2 x + 1 -\left( x + 16 \right)$

$0 > x^{2} + x - 15$

Now we can use the quadratic formula.

Recall the quadratic formula.

$\frac{ -b \pm \sqrt{ b ^2 -4 \cdot a \cdot c }}{ 2 \cdot a }$

Where $\uptext{a}$ , $\uptext{b}$ , and $\uptext{c}$ , correspond to coefficients in the quadratic equation.

$a x^{2} + b x + c = 0$

In this case a = 1 , b = 1 , and c = -15 .

Now plug these values into the quadratic equation, and we get.

x = 3.4

x = -4.4

Now since we are dealing with an inequality, we put the least value on the left side, and the greatest value on the right. It will look like the following.

-4.4 < x < 3.4

Report an Error

Example Question #192 : Equations / Inequalities

Solve the following inequality for $\uptext{x}$ , round your answer to the nearest tenth.

$\sqrt{x + 34} > x + 1$

Possible Answers:

x = 5.3

-6.3 < x < 5.3

$-6.3 \leq x$

5.3 < x < -6.3

5.3 < x

Correct answer:

-6.3 < x < 5.3

Explanation:

The first step is to square each side of the inequality.

$\left( \sqrt{x + 34} \right)^2> \left( x + 1 \right)^2$

Now simplify each side.

$x + 34 > x^{2} + 2 x + 1$

Now subtract the left side of the inequality to make it zero, so that we can use the quadratic formula.

$x + 34 -\left( x + 34 \right)> x^{2} + 2 x + 1 -\left( x + 34 \right)$

$0 > x^{2} + x - 33$

Now we can use the quadratic formula.

Recall the quadratic formula.

$\frac{ -b \pm \sqrt{ b ^2 -4 \cdot a \cdot c }}{ 2 \cdot a }$

Where $\uptext{a}$ , $\uptext{b}$ , and $\uptext{c}$ , correspond to coefficients in the quadratic equation.

$a x^{2} + b x + c = 0$

In this case a = 1 , b = 1 , and c = -33 .

Now plug these values into the quadratic equation, and we get.

x = 5.3

x = -6.3

Now since we are dealing with an inequality, we put the least value on the left side, and the greatest value on the right. It will look like the following.

-6.3 < x < 5.3

Report an Error

Example Question #193 : Equations / Inequalities

Solve the following inequality for $\uptext{x}$ , round your answer to the nearest tenth.

$\sqrt{x + 19} > x + 1$

Possible Answers:

x = 3.8

3.8 < x < -4.8

-4.8 < x < 3.8

3.8 < x

$-4.8 \leq x$

Correct answer:

-4.8 < x < 3.8

Explanation:

The first step is to square each side of the inequality.

$\left( \sqrt{x + 19} \right)^2> \left( x + 1 \right)^2$

Now simplify each side.

$x + 19 > x^{2} + 2 x + 1$

Now subtract the left side of the inequality to make it zero, so that we can use the quadratic formula.

$x + 19 -\left( x + 19 \right)> x^{2} + 2 x + 1 -\left( x + 19 \right)$

$0 > x^{2} + x - 18$

Now we can use the quadratic formula.

Recall the quadratic formula.

$\frac{ -b \pm \sqrt{ b ^2 -4 \cdot a \cdot c }}{ 2 \cdot a }$

Where $\uptext{a}$ , $\uptext{b}$ , and $\uptext{c}$ , correspond to coefficients in the quadratic equation.

$a x^{2} + b x + c = 0$

In this case a = 1 , b = 1 , and c = -18 .

Now plug these values into the quadratic equation, and we get.

x = 3.8

x = -4.8

Now since we are dealing with an inequality, we put the least value on the left side, and the greatest value on the right. It will look like the following.

-4.8 < x < 3.8

Report an Error

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PSAT Math : Systems of Inequalities

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #1 : Solve One Variable Linear Equations And Inequalities: Ccss.Math.Content.Hsa Rei.B.3

Example Question #491 : High School: Algebra

Example Question #191 : Equations / Inequalities

Example Question #192 : Equations / Inequalities

Example Question #193 : Equations / Inequalities

All PSAT Math Resources

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