Grade 9 · Algebra & functions

Linear inequality system solutions practice

Linear inequality system solutions is a grade 9 math skill aligned to Common Core standard HSA.REI.D.12: graph the solutions to a linear inequality in two variables as a half-plane, and graph the solution set to a system of linear inequalities as the intersection of half-planes. Below are 8 practice questions with answers and step-by-step explanations, drawn from the 10 linear inequality system solutions problems our math games drill.

CCSS HSA.REI.D.1210 questions in the bank
Sample questions

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Question 1easy

{y>3x5 yx+4\begin{cases} y > 3x - 5 \ y \le x + 4 \end{cases}
The point
(x,y)(x, y) is a solution to the system of inequalities in the coordinate plane. If x=2x = 2, which of the following could be a value of yy?

Substitute x=2x = 2. From y>3x5y > 3x - 5, y>1y > 1. From yx+4y \le x + 4, y6y \le 6. The value y=4y = 4 satisfies 1<461 < 4 \le 6.

Question 2easy

{yx+3 y<2x+7\begin{cases} y \ge -x + 3 \ y < 2x + 7 \end{cases}
The point
(x,y)(x, y) is a solution to the system of inequalities in the coordinate plane. If x=1x = 1, which of the following could be a value of yy?

Substitute x=1x = 1. From yx+3y \ge -x + 3, y2y \ge 2. From y<2x+7y < 2x + 7, y<9y < 9. The value y=5y = 5 satisfies 25<92 \le 5 < 9.

Question 3easy

{yx+10 y>x1\begin{cases} y \le -x + 10 \ y > x - 1 \end{cases}
The point
(x,y)(x, y) is a solution to the system of inequalities in the coordinate plane. If x=4x = 4, which of the following could be a value of yy?

Substitute x=4x = 4. From yx+10y \le -x + 10, y6y \le 6. From y>x1y > x - 1, y>3y > 3. The value y=5y = 5 satisfies 3<563 < 5 \le 6.

Question 4easy

{y2x+1 yx+5\begin{cases} y \ge 2x + 1 \ y \le x + 5 \end{cases}
The point
(x,y)(x, y) is a solution to the system of inequalities in the coordinate plane. If x=0x = 0, which of the following could be a value of yy?

Substitute x=0x = 0. From y2x+1y \ge 2x + 1, y1y \ge 1. From yx+5y \le x + 5, y5y \le 5. The value y=3y = 3 satisfies 1351 \le 3 \le 5.

Question 5medium

{x+y10 xy2\begin{cases} x + y \le 10 \ x \ge y - 2 \end{cases}
The point
(x,y)(x, y) is a solution to the system of inequalities in the coordinate plane. If y=6y = 6, which of the following could be a value of xx?

Substitute y=6y = 6. From x+y10x + y \le 10, x4x \le 4. From xy2x \ge y - 2, x4x \ge 4. The only listed value that satisfies both is x=4x = 4.

Question 6medium

{2xy4 x+y7\begin{cases} 2x - y \ge 4 \ x + y \le 7 \end{cases}
The point
(x,y)(x, y) is a solution to the system of inequalities in the coordinate plane. If y=2y = 2, which of the following could be a value of xx?

Substitute y=2y = 2. From 2xy42x - y \ge 4, 2x242x - 2 \ge 4, so x3x \ge 3. From x+y7x + y \le 7, x5x \le 5. The value x=4x = 4 satisfies 3453 \le 4 \le 5.

Question 7medium

{x2y<3 x+y>6\begin{cases} x - 2y < -3 \ x + y > 6 \end{cases}
The point
(x,y)(x, y) is a solution to the system of inequalities in the coordinate plane. If y=5y = 5, which of the following could be a value of xx?

Substitute y=5y = 5. From x2y<3x - 2y < -3, x10<3x - 10 < -3, so x<7x < 7. From x+y>6x + y > 6, x+5>6x + 5 > 6, so x>1x > 1. The value x=4x = 4 satisfies 1<4<71 < 4 < 7.

Question 8medium

{y2x+1 yx+9\begin{cases} y \ge 2x + 1 \ y \le x + 9 \end{cases}
The point
(x,y)(x, y) is a solution to the system of inequalities in the coordinate plane. If y=8y = 8, which of the following could be a value of xx?

Substitute y=8y = 8. From y2x+1y \ge 2x + 1, 82x+18 \ge 2x + 1, so x72x \le \dfrac{7}{2}. From yx+9y \le x + 9, 8x+98 \le x + 9, so x1x \ge -1. The value x=2x = 2 satisfies both inequalities.

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